Solve each equation. x=log⁡775x = \$$log_7$$ \sqrt[5]{7}

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

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Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

문제 24

5장, 문제 24

Solve each equation. $x = \log_{7} \sqrt[5]{7}$

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Recognize that the equation is given as $x = \log_{7} \sqrt[5]{7}$, where the logarithm has base 7 and the argument is the fifth root of 7.

Rewrite the fifth root of 7 using exponents: $\sqrt[5]{7} = 7^{\frac{1}{5}}$.

Substitute this back into the logarithm: $x = \log_{7} \left(7^{\frac{1}{5}}\right)$.

Use the logarithm power rule, which states $\log_{a} (b^{c}) = c \cdot \log_{a} b$, to simplify: $x = \frac{1}{5} \cdot \log_{7} 7$.

Since $\log_{7} 7 = 1$ (because the log base and the argument are the same), simplify to find $x = \frac{1}{5}$.

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Logarithms and Their Properties

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log₇(7) = 1 because 7¹ = 7. Understanding how to interpret and manipulate logarithms is essential for solving equations involving them.

Radicals and Exponents

A radical such as the fifth root (⁵√7) can be expressed as an exponent: 7^(1/5). Converting radicals to fractional exponents simplifies working with logarithms and helps in applying logarithmic properties effectively.

Change of Base and Simplification

Using the property log_b(a^c) = c * log_b(a), you can simplify logarithmic expressions by bringing exponents in front as multipliers. This property is key to solving equations like x = log₇(⁵√7) by rewriting the root as an exponent and simplifying.

Solve each equation. x=log775x = \(\log\)_7 \(\sqrt\)[5]{7}