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Express log6 216 = 3 in exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithmic and Exponential Relationships"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to convert a logarithmic equation to its equivalent exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic form: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponential form: "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the base ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"), the argument ("},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"), and the result ("},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":") in the logarithmic equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}},{"type":"text","text":" means "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":" raised to the power "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" equals "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the equivalent exponential equation using the identified values."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"6^3 = 216"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We converted the logarithmic equation to exponential form by raising the base to the exponent."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Express log3 81 = 4 in exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithmic and Exponential Relationships"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert logarithmic equations to exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic form: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponential form: "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the base ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"), argument ("},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"), and exponent ("},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":") in the logarithmic equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the conversion: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}},{"type":"text","text":" is equivalent to "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the exponential equation using these values."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"3^4 = 81"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The base 3 raised to the power 4 equals 81, matching the logarithmic statement."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Express log3 (1/9) = -2 in exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithmic and Exponential Relationships"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert logarithmic equations with negative exponents to exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic form: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponential form: "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the base ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"), argument ("},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"), and exponent ("},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":") in the logarithmic equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that a negative exponent means the reciprocal: "},{"type":"inlineMath","attrs":{"latex":"b^{-2} = \\frac{1}{b^2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the exponential equation using these values."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"3^{-2} = \\frac{1}{9}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Negative exponents indicate reciprocals, so "},{"type":"inlineMath","attrs":{"latex":"3^{-2}"}},{"type":"text","text":" equals "},{"type":"inlineMath","attrs":{"latex":"1/9"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Express log64 8 = 1/2 in exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithmic and Exponential Relationships"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert logarithmic equations with fractional exponents to exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic form: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponential form: "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the base ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"), argument ("},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"), and exponent ("},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":") in the logarithmic equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that a fractional exponent represents a root: "},{"type":"inlineMath","attrs":{"latex":"b^{1/2} = \\sqrt{b}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the exponential equation using these values."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"64^{1/2} = 8"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The exponent "},{"type":"inlineMath","attrs":{"latex":"1/2"}},{"type":"text","text":" means the square root, so "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{64} = 8"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Express log7 = 0.845 in exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithmic and Exponential Relationships"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert logarithmic equations with decimal exponents to exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic form: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponential form: "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the base ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":") and exponent ("},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":") in the logarithmic equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the conversion: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}},{"type":"text","text":" is equivalent to "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the exponential equation using these values."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"7^{0.845} = a"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The base 7 raised to the power 0.845 equals the argument "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Express log1/2 32 = -5 in exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithmic and Exponential Relationships"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert logarithmic equations with negative exponents and fractional bases to exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic form: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponential form: "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the base ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"), argument ("},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"), and exponent ("},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":") in the logarithmic equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that a negative exponent means the reciprocal: "},{"type":"inlineMath","attrs":{"latex":"b^{-5} = \\frac{1}{b^5}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the exponential equation using these values."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"(\\frac{1}{2})^{-5} = 32"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Negative exponents indicate reciprocals, and raising "},{"type":"inlineMath","attrs":{"latex":"1/2"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"-5"}},{"type":"text","text":" gives 32."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Express 27 = 128 in logarithmic form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Exponential and Logarithmic Relationships"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert an exponential equation to logarithmic form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponential form: "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic form: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the base ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"), exponent ("},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":"), and result ("},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":") in the exponential equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"b^c = a"}},{"type":"text","text":" is equivalent to "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the logarithmic equation using these values."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\log_2 128 = 7"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The logarithm base 2 of 128 equals 7."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. 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Evaluate log5 57 without a calculator."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Logarithms"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to estimate logarithms when the argument is not a simple power of the base."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic form: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If "},{"type":"inlineMath","attrs":{"latex":"a = b^c"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the closest powers of 5 to 57 (e.g., "},{"type":"inlineMath","attrs":{"latex":"5^2 = 25"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"5^3 = 125"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Estimate where 57 falls between these powers."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the logarithmic equation and consider how to estimate "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\log_5 57 \\approx 2.8"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"57 is between "},{"type":"inlineMath","attrs":{"latex":"5^2"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"5^3"}},{"type":"text","text":", so the logarithm is between 2 and 3."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. Evaluate "},{"type":"inlineMath","attrs":{"latex":"4\\log_4 6"}},{"type":"text","marks":[{"type":"bold"}],"text":" without a calculator."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Properties of Logarithms"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the power rule for logarithms."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Power rule: "},{"type":"inlineMath","attrs":{"latex":"k \\log_b a = \\log_b a^k"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the power rule to rewrite "},{"type":"inlineMath","attrs":{"latex":"4\\log_4 6"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\log_4 6^4"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"6^4"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the logarithmic equation "},{"type":"inlineMath","attrs":{"latex":"\\log_4 6^4"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"4\\log_4 6 = \\log_4 1296"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using the power rule, "},{"type":"inlineMath","attrs":{"latex":"4\\log_4 6"}},{"type":"text","text":" is equivalent to "},{"type":"inlineMath","attrs":{"latex":"\\log_4 1296"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q18. Evaluate log25 5 without a calculator."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Logarithms"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to evaluate logarithms with a base that is a power of the argument."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic form: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If "},{"type":"inlineMath","attrs":{"latex":"a = b^c"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = c"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express 25 as a power of 5 ("},{"type":"inlineMath","attrs":{"latex":"25 = 5^2"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\log_{25} 5 = c"}},{"type":"text","text":" means "},{"type":"inlineMath","attrs":{"latex":"25^c = 5"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the equation and solve for "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\log_{25} 5 = \\frac{1}{2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Because "},{"type":"inlineMath","attrs":{"latex":"25^{1/2} = 5"}},{"type":"text","text":", the logarithm equals 1/2."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. Use a calculator to evaluate log4 18 to four decimal places."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Logarithms with Calculators"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the change of base formula to evaluate logarithms with non-standard bases."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Change of base formula: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = \\frac{\\log a}{\\log b}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the change of base formula: "},{"type":"inlineMath","attrs":{"latex":"\\log_4 18 = \\frac{\\log 18}{\\log 4}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use your calculator to find "},{"type":"inlineMath","attrs":{"latex":"\\log 18"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\log 4"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide the two values to get the result to four decimal places."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\log_4 18 \\approx 2.0841"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using the change of base formula and calculator, the result is approximately 2.0841."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q20. Use a calculator to evaluate log1/2 23 to four decimal places."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Logarithms with Calculators"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the change of base formula for logarithms with fractional bases."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Change of base formula: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = \\frac{\\log a}{\\log b}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the change of base formula: "},{"type":"inlineMath","attrs":{"latex":"\\log_{1/2} 23 = \\frac{\\log 23}{\\log (1/2)}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use your calculator to find "},{"type":"inlineMath","attrs":{"latex":"\\log 23"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\log (1/2)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide the two values to get the result to four decimal places."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\log_{1/2} 23 \\approx -4.5237"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using the change of base formula and calculator, the result is approximately -4.5237."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q21. Use a calculator to evaluate log\\pi 57 to four decimal places."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Logarithms with Calculators"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the change of base formula for logarithms with irrational bases."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Change of base formula: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = \\frac{\\log a}{\\log b}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the change of base formula: "},{"type":"inlineMath","attrs":{"latex":"\\log_{\\pi} 57 = \\frac{\\log 57}{\\log \\pi}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use your calculator to find "},{"type":"inlineMath","attrs":{"latex":"\\log 57"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\log \\pi"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide the two values to get the result to four decimal places."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\log_{\\pi} 57 \\approx 4.0472"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using the change of base formula and calculator, the result is approximately 4.0472."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q22. Use a calculator to evaluate log0.8 16 to four decimal places."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Logarithms with Calculators"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the change of base formula for logarithms with bases less than 1."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Change of base formula: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = \\frac{\\log a}{\\log b}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the change of base formula: "},{"type":"inlineMath","attrs":{"latex":"\\log_{0.8} 16 = \\frac{\\log 16}{\\log 0.8}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use your calculator to find "},{"type":"inlineMath","attrs":{"latex":"\\log 16"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\log 0.8"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide the two values to get the result to four decimal places."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\log_{0.8} 16 \\approx -13.2877"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using the change of base formula and calculator, the result is approximately -13.2877."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q23. Use a calculator to evaluate log1 36 to four decimal places."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Logarithms with Calculators"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the change of base formula for logarithms with base 1, which is undefined."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Change of base formula: "},{"type":"inlineMath","attrs":{"latex":"\\log_b a = \\frac{\\log a}{\\log b}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that logarithms with base 1 are undefined because "},{"type":"inlineMath","attrs":{"latex":"1^c = a"}},{"type":"text","text":" only works for "},{"type":"inlineMath","attrs":{"latex":"a = 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Attempting to use the change of base formula will result in division by zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recognize that "},{"type":"inlineMath","attrs":{"latex":"\\log_1 36"}},{"type":"text","text":" is not defined."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\log_1 36"}},{"type":"text","marks":[{"type":"bold"}],"text":" is undefined"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithms with base 1 are not defined for any value other than 1."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q24. Use a calculator to evaluate log9 (0.85) to four decimal places."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Logarithms with Calculators"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the change of base formula for logarithms with bases greater than 1 and arguments less than 1."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[