Problem 28
Question
Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{9}$$
Step-by-Step Solution
Verified Answer
-2
1Step 1: 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 \(\frac {1} {9}\).
2Step 2: 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 \(-2*log_3{3}\).
3Step 3: 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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