Textbook Question
Write each equation in its equivalent exponential form. log3 81 = y
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6. Exponential & Logarithmic Functions
Introduction to Logarithms
1:35 minutesProblem 19
Textbook Question
Write each equation in its equivalent logarithmic form. 7y = 200
Verified step by step guidance1
Identify the given exponential equation: \$7^y = 200$.
Recall the definition of logarithms: if \(a^x = b\), then the equivalent logarithmic form is \(\log_a b = x\).
In this problem, the base \(a\) is 7, the exponent \(x\) is \(y\), and the result \(b\) is 200.
Rewrite the equation \$7^y = 200$ in logarithmic form using the definition: \(\log_7 200 = y\).
This expresses the original exponential equation as a logarithmic equation, which is the equivalent form requested.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential and Logarithmic Forms
Exponential and logarithmic forms are two ways to express the same relationship. An equation like a^y = b can be rewritten in logarithmic form as log_a(b) = y, where the base a remains the same. Understanding this equivalence is essential for converting between forms.
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Solving Logarithmic Equations
Definition of a Logarithm
A logarithm answers the question: to what power must the base be raised to produce a given number? For example, log_a(b) = y means a raised to y equals b. This definition is fundamental for rewriting exponential equations as logarithms.
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Properties of Logarithms
Logarithms have specific properties, such as the base must be positive and not equal to 1, and the argument must be positive. These properties ensure the logarithmic expression is valid and help in correctly rewriting and solving equations.
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