/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 110 Solve. $$ \log _{8} x=\frac{... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 110

Solve. $$ \log _{8} x=\frac{2}{3} $$

Short Answer

Expert verified
x = 4

Step by step solution

01

Understand the Logarithmic Equation

Recognize that the equation \( \log _{8} x=\frac{2}{3} \) can be rewritten in its exponential form: \( x = 8^{\frac{2}{3}} \).
02

Convert to Exponential Form

Rewrite the logarithmic equation in exponential form: \( x = 8^{\frac{2}{3}} \).
03

Simplify the Exponent

Recognize that \( 8 = 2^3 \). Substituting this into the equation gives: \( 8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} \).
04

Simplify Using Properties of Exponents

Use the property of exponents \( (a^m)^n = a^{mn} \) to simplify: \( (2^3)^{\frac{2}{3}} = 2^{3 \cdot \frac{2}{3}} = 2^2 \).
05

Calculate the Result

Evaluate the exponent: \( 2^2 = 4 \). Hence, \( x = 4 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Exponents
Simplifying exponents involves using the properties of exponents to break down complex expressions. In the given problem, the exponent form of 8 is used: \( 8 = 2^3 \). Substituting this, we get \( (2^3)^{\frac{2}{3}} \). Using the property \( (a^m)^n = a^{mn} \), this simplifies to \( 2^{3\cdot\frac{2}{3}} = 2^2 \).
Finally, \( 2^2 \) simplifies to 4. So, when you are faced with such equations, recognizing ways to simplify using known exponent properties is key.

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