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

Verified step by step guidance

1. 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.
2. Rewrite the fifth root of 7 using exponents: \(\sqrt[5]{7} = 7^{\frac{1}{5}}\).
3. Substitute this back into the logarithm: \(x = \log_{7} \left(7^{\frac{1}{5}}\right)\).
4. Use the logarithm power rule, which states \(\log_{a} (b^{c}) = c \cdot \log_{a} b\), to simplify: \(x = \frac{1}{5} \cdot \log_{7} 7\).
5. Since \(\log_{7} 7 = 1\) (because the log base and the argument are the same), simplify to find \(x = \frac{1}{5}\).

Key Concepts

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.