Problem 32
Question
Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{\sqrt{3}}$$
Step-by-Step Solution
Verified Answer
-1/2
1Step 1: Converting the square root to exponent
Rewrite the \(\sqrt{3}\) as 3 raised to the power of 1/2, so \(\frac{1}{\sqrt{3}}\) becomes \(\frac{1}{3^{1/2}}\) or \(3^{-1/2}\). Thus, \(\log _{3} \frac{1}{\sqrt{3}}\) becomes \(\log _{3} 3^{-1/2}\)
2Step 2: Using the power rule of logarithm
The power rule, \(\log_b a^n = n \log_b a\), allows us to bring the exponent in front, so \(\log_3 3^{-1/2} = -1/2 * \log_3 3\).
3Step 3: Solving for the Logarithm of the same base
The property of logarithms states that \(\log_b b = 1\). Therefore, \(-1/2 * \log_3 3\) is equal to \(-1/2 * 1\).
Other exercises in this chapter
Problem 31
Evaluate each expression without using a calculator. $$\log _{2} \frac{1}{\sqrt{2}}$$
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