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Chapter 12: Problem 25

Evaluate each expression without using a calculator. $$\log _{5} \frac{1}{5}$$

Short Answer

Expert verified
The value of \(\log _{5} \frac{1}{5}\) is -1

Step by step solution

01

Understand the logarithm definition

Based on the definition of a logarithm, \(\log _{b} a=n\) if and only if \(b^{n}=a\). It implies that the base \(b\), raised to the power \(n\), equals \(a\).
02

Apply the logarithm definition to the given expression

Applying the definition to the given expression would mean finding a number 'n' such that when 5 (the base) is raised to the power 'n' equals \(\frac{1}{5}\). This can be written as \(5^n = \frac{1}{5}\)
03

Transform the expression

It is evident that \( \frac{1}{5}\) can be written as \(5^{-1}\), therefore, \(5^n = 5^{-1}\).
04

Compare the exponentials

Using the property of equality of expressions, n should be equal to -1. This is because the base of both sides is the same (which is 5), thus their exponents must equal each other. Hence \(n = -1\)

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)

Solve the system: $$\left\\{\begin{aligned}2 x &=11-5 y \\\3 x-2 y &=-12\end{aligned}\right.$$ (Section 4.3, Example 4)

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\ln x, g(x)=\ln x+3$$

Solve: $$\sqrt{2 x-1}-\sqrt{x-1}=1$$

Solve each equation. $$\log _{2}(x-6)+\log _{2}(x-4)-\log _{2} x=2$$
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