Solve each equation. $x={\log }_8\sqrt[4]{8}$

Verified step by step guidance

1. Recognize the given equation: \(x = \log_{8} \sqrt[4]{8}\). The goal is to solve for \(x\).
2. Rewrite the fourth root as a fractional exponent: \(\sqrt[4]{8} = 8^{\frac{1}{4}}\).
3. Substitute this back into the logarithm: \(x = \log_{8} \left(8^{\frac{1}{4}}\right)\).
4. Use the logarithm power rule: \(\log_{a} (b^{c}) = c \cdot \log_{a} b\). So, \(x = \frac{1}{4} \cdot \log_{8} 8\).
5. Since \(\log_{8} 8 = 1\) (because the base and the argument are the same), simplify to find \(x = \frac{1}{4}\).

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, log₈(8) = 1 because 8¹ = 8. Understanding how to interpret and manipulate logarithms is essential for solving equations involving them.

Radicals and Fractional Exponents

Radicals like the fourth root (∜) can be expressed as fractional exponents, such as ∜8 = 8^(1/4). Converting radicals to exponents simplifies calculations and helps in applying logarithmic rules effectively.

Change of Base and Simplification Techniques

Simplifying logarithmic expressions often involves rewriting numbers with common bases or using properties like log_b(a^c) = c·log_b(a). Recognizing these allows for easier evaluation of expressions like log₈(8^(1/4)) by bringing the exponent out front.