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Chapter 1: Problem 43

Solve the following equations. $$\log _{8} x=\frac{1}{3}$$

Short Answer

Expert verified
Answer: The value of x in the given logarithmic equation is $$x = 2$$.

Step by step solution

01

Understand the given equation

The given equation is: $$\log_{8} x = \frac{1}{3}.$$ It is a logarithmic equation, where the base is 8, and we need to find the value of x.
02

Rewrite the equation in exponential form

We can rewrite the given logarithmic equation in exponential form using the basic property of logarithms: $$b^{\log_{b}{a}} = a.$$ In our case, we have: $$8^{\frac{1}{3}} = x.$$
03

Solve for x

To solve for x, we have to find the cube root of 8. Cube root means finding the number that, when multiplied by itself three times, yields the number inside the root symbol. In this case, the cube root of 8 is 2 because: $$2 * 2 * 2 = 8.$$ So, $$x = 8^{\frac{1}{3}} = 2.$$
04

Write the final answer

The solution to the given logarithmic equation is: $$x = 2.$$

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