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Chapter 13: Problem 16

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{8} 10^{4}$$

Short Answer

Expert verified
\( \log _{8} 10^{4} = \frac{4}{\log_{10}{8}} \)

Step by step solution

01

Analyze the given expression

We have the expression \(\log _{8} 10^{4}\). We will first simplify the expression by using the power property of logarithms which states that \(\log_b{a^n}=n \log_b{a}\).
02

Apply the power property of logarithm

Using the power property of logarithms, we have \[ \log _{8} 10^{4} = 4 \log _{8} 10 \]
03

Use change of base formula to convert to common logarithm

We will convert the given expression into a common logarithm (with base 10) using the change of base formula. The change of base formula states that, given \(\log_{b}{a}=\frac{\log_{c}{a}}{\log_{c}{b}}\), we can rewrite the logarithm with new base c. Here we will use the common logarithm (base 10). \[ 4 \log_{8}{10} = 4 \frac{\log_{10}{10}}{\log_{10}{8}} \]
04

Simplify the common logarithm expression

Now we'll simplify the common logarithm expression. \[ 4 \frac{\log_{10}{10}}{\log_{10}{8}} = 4 \frac{1}{\log_{10}{8}} \]
05

Final expression

After simplifying the expression from the previous steps, the final expression for the given exercise is \[ \log _{8} 10^{4} = \frac{4}{\log_{10}{8}} \]

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Most popular questions from this chapter

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