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Chapter 12: Problem 16

Simplify. $$ \log _{4} \frac{1}{4} $$

Short Answer

Expert verified
\(\log_{4} \frac{1}{4} = -1\).

Step by step solution

01

Understand the logarithm properties

Recall that for any logarithm \(\log_b a\), it is the exponent to which the base \(b\) must be raised to get the value \(a\).
02

Set up the equation

To simplify \(\log_{4} \frac{1}{4}\), let \(x = \log_{4} \frac{1}{4}\). This implies that \(4^x = \frac{1}{4}\).
03

Rewrite the fraction

Rewrite \(\frac{1}{4}\) as \(4^{-1}\). So now, you have \(4^x = 4^{-1}\).
04

Equate the exponents

Since the bases are the same, you can equate the exponents: \(x = -1\).
05

Write the final simplified form

Thus, \(\log_{4} \frac{1}{4} = -1\).

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Key Concepts

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

Logarithm Properties
Logarithms can seem tricky at first, but they follow simple, predictable rules. One of the most important properties is that a logarithm is an exponent.
When you see \(\log_b a\)\, it means: \

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Most popular questions from this chapter

Solve. $$ \log _{4}(3 x-2)=2 $$

Find a formula for converting natural logarithms to common logarithms.

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b}(3 b)$$

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