Problem 32 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

-0.5

Step 1: Understanding the logarithmic expression

Logarithms are ways of expressing numbers, and the expression \(\log _{3} \frac{1}{\sqrt{3}}\) means to what power the base (in this case 3) must be raised to get the number inside the log (in this case \(\frac{1}{\sqrt{3}}\)).

Step 2: Express the number inside the log as exponent of 3

Let's express \(\frac{1}{\sqrt{3}}\) as an exponent of 3. We all know that, \(3^2=9\) and \(3^{-1}=\frac{1}{3}\). So, as per the rules of square roots, \(3^{0.5}=\sqrt{3}\). Thus, \(\frac{1}{\sqrt{3}} = \frac{1}{3^{0.5}} = 3^{-0.5}\). This means our expression can be rewritten as \(\log _{3} {3^{-0.5}}\).

Step 3: Simplify the expression

The general property of logarithm, \(\log_b{b^x} = x\), can be applied now. So, \(\log _{3} {3^{-0.5}} = -0.5\).