Write each equation in its equivalent logarithmic form. 13² = x

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Identify the given exponential equation: \(13^2 = x\).

Recall the definition of logarithms: if \(a^b = c\), then the equivalent logarithmic form is \(\log_a c = b\).

In this problem, the base \(a\) is 13, the exponent \(b\) is 2, and the result \(c\) is \(x\).

Apply the logarithmic form to rewrite the equation as \(\log_{13} x = 2\).

This is the equivalent logarithmic form of the given exponential equation.

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Key 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^b = c can be rewritten as log_a(c) = b, where the logarithm answers the question: to what power must the base a be raised to get c?

Definition of a Logarithm

A logarithm log_a(x) is the inverse operation of exponentiation, meaning it finds the exponent to which the base a must be raised to produce x. This definition is fundamental for converting between exponential and logarithmic equations.

Properties of Logarithms

Understanding properties such as the base must be positive and not equal to 1, and the argument must be positive, is essential. These properties ensure the logarithmic form is valid and help in correctly rewriting exponential equations.

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