Problem 32 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

\(-0.5\)

Step 1: Understanding Logarithmic Expression

The expression \(\log_3 \frac{1}{\sqrt{3}}\) can be broken down with the understanding that the logarithm base 3 of 1 divided by the square root of 3 is the power to which 3 must be raised to get \(\frac{1}{\sqrt{3}}\).

Step 2: Simplifying Numerator

Notice that \(\frac{1}{\sqrt{3}}\) can be simplified. We have to rationalize the denominator by multiplying both numerator and denominator by \(\sqrt{3}\), which yields \(\frac{\sqrt{3}}{3}\). Now our expression becomes \(\log_3 \frac{\sqrt{3}}{3}\).

Step 3: Using Properties of Logarithms

The expression can now be rewritten into subtraction form utilizing logarithmic rules. According to the rules, \(\log_b \frac{m}{n}= \log_b m - \log_b n\). It then becomes \(\log_3 {\sqrt{3}} - \log_3 3\).

Step 4: Calculate Individual Logarithms

Finally, the log base 3 of square root of 3 is equal to \(0.5\), because 3 raised to power \(0.5\) equals to square root of 3. And log base 3 of 3 is equal to \(1\), because 3 raised to power 1 is 3. Our final expression is therefore \(0.5 - 1\).

Step 5: Final Calculation

Subtracting 1 from 0.5 results to \(-0.5\). That's the final answer.