Solve each equation. x=3log⁡38x = 3 \$$log_3 8$$

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

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Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

문제 25

5장, 문제 25

Solve each equation. $x = 3 \log_{3} 8$

검증된 단계별 안내

Recognize that the equation is given as $x = 3^{\log_3 8}$, where the base of the exponent and the base of the logarithm are the same (both 3).

Recall the logarithmic identity: $a^{\log_a b} = b$. This means that when the base of the exponent and the logarithm match, the expression simplifies directly to the argument of the logarithm.

Apply this identity to simplify $3^{\log_3 8}$ to just 8.

Therefore, the value of $x$ is equal to 8.

Conclude that the solution to the equation is $x = 8$.

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주요 개념

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Logarithmic and Exponential Functions

Logarithmic functions are the inverses of exponential functions. Understanding how these two relate helps in simplifying expressions like 3^log3(8), where the base of the exponent and the logarithm are the same.

Properties of Logarithms

Key properties such as log_b(b^x) = x and b^{log_b(x)} = x allow simplification of expressions involving logs and exponents with the same base. These properties are essential for solving equations like x = 3^{log_3(8)}.

Evaluating Expressions with Matching Bases

When the base of the exponent matches the base of the logarithm, the expression simplifies directly to the argument of the logarithm. For example, 3^{log_3(8)} simplifies to 8, which is crucial for solving the given equation.

Solve each equation. x=3log38x = 3 \(\log\)_3 8