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Chapter 12: Problem 28

Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{9}$$

Short Answer

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Step by step solution

01

Recognize the problem as a logarithm

In a logarithmic function of the form \(\log_b {a}\) , b is the base, and a is the number for which we want to find the logarithm. In this problem, we are asked to find \(\log_3{\frac{1}{9}}\), i.e., the power to which we must raise 3 to obtain \(\frac {1} {9}\).
02

Use the power rule of logarithms

We can write the fraction inside the logarithm as a power of 3, i.e., \(3^{-2} = \frac{1}{9}\). Using the power rule of logarithms, this simplifies the expression as \(-2*log_3{3}\).
03

Simplify using the rule \(\log{_b}{b} = 1\)

Using the identity that the log of the base to itself is equal to 1, the expression simplifies to \(-2*1\).

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Most popular questions from this chapter

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; \quad 1-\frac{1}{2}+\frac{1}{3} ; \quad 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}, \ldots\) Describe what you observe.

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

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

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$3^{x+1}=9$$

Describe the change-of-base property and give an example.
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