Textbook Question
Solve each equation. Give solutions in exact form. log2 [(2x + 8)(x + 4)] = 5
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6. Exponential & Logarithmic Functions
Solving Exponential and Logarithmic Equations
1:42 minutesProblem 59
Textbook Question
Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log4(3x+2)=3
Verified step by step guidance1
Recall the definition of logarithm: if \(\log_{a}(b) = c\), then it is equivalent to the exponential form \(a^{c} = b\).
Rewrite the given equation \(\log_{4}(3x + 2) = 3\) in exponential form: \$4^{3} = 3x + 2$.
Calculate \$4^{3}\( (but do not finalize the numeric value) and set up the equation: \)64 = 3x + 2$.
Solve for \(x\) by isolating it: subtract 2 from both sides to get \$64 - 2 = 3x\(, then divide both sides by 3 to find \)x = \frac{62}{3}$.
Check the domain restriction: the argument of the logarithm, \$3x + 2\(, must be greater than 0. Substitute \)x = \frac{62}{3}\( to verify \)3\left(\frac{62}{3}\right) + 2 > 0$ to ensure the solution is valid.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Understanding the properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows you to rewrite logarithmic equations in exponential form to solve for the variable. Recognizing how to manipulate logs helps simplify and solve the given equation.
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Domain of Logarithmic Functions
The domain of a logarithmic function includes all values for which the argument (inside the log) is positive. For log_4(3x+2), the expression 3x+2 must be greater than zero. Identifying and applying domain restrictions ensures that solutions are valid and prevents extraneous answers.
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Exact and Approximate Solutions
After solving the equation exactly, it is often necessary to provide a decimal approximation. Using a calculator to find decimal values correct to two decimal places helps interpret the solution practically. Distinguishing between exact and approximate answers is important in many algebra problems.
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