Solve each equation. log4 x = 3

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

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Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

문제 31

5장, 문제 31

Solve each equation. log₄ x = 3

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Recognize that the equation is given in logarithmic form: $\log_{4} x = 3$.

Recall the definition of a logarithm: $\log_{a} b = c$ means $a^{c} = b$.

Rewrite the equation $\log_{4} x = 3$ in its equivalent exponential form: $4^{3} = x$.

Calculate the value of \$4^{3}$ by multiplying 4 by itself three times (do not provide the final number here).

Conclude that the solution for $x$ is the value obtained from \$4^{3}$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For log₄ x = 3, it means 4 raised to the power 3 equals x. Understanding this definition allows you to rewrite logarithmic equations in exponential form.

Converting Logarithmic Equations to Exponential Form

To solve log₄ x = 3, rewrite it as an exponential equation: 4³ = x. This conversion simplifies solving for x by removing the logarithm and using basic exponentiation, making it easier to find the solution.

Properties of Exponents

Knowing how to compute powers, such as 4³, is essential. Exponentiation involves multiplying the base by itself as many times as the exponent indicates. Here, 4³ = 4 × 4 × 4 = 64, which gives the solution to the equation.