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Chapter 3: Problem 30

Find the exact value of the logarithmic expression without using a calculator. (If this is not possible, state the reason.) $$\log _{5} \frac{1}{125}$$

Short Answer

Expert verified
-3 is the exact value of the logarithmic expression given without the use of a calculator.

Step by step solution

01

Rewrite the fraction as a power of 5

Firstly, the fraction \(\frac{1}{125}\) can be written as the negative power of 5, \( 5^{-3} \), because \( 5^3 = 125 \). Hence, \(\frac{1}{125} = 5^{-3} \).
02

Use the law of logarithms

Then, we substitute \( 5^{-3} \) into the given logarithmic expression. Hence, we get \( \log_{5}{5^{-3}} \). Logarithms have a unique property such that \( \log_{b}{b^x} = x \). So it simplifies to -3.

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