Problem 28 - College Algebra

Question

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

Step-by-Step Solution

Answer

The value of the logarithmic expression \(log_3 (1/9)\) is -2.

Step 1: Understanding the given expression

The given log expression is \(log_3 (1/9)\) . Here, the base of the logarithm is 3 and the argument of the logarithm is \(1/9\).

Step 2: Express the fraction as a power of the base

We can express \(1/9\) as \(3^{-2}\) . This is because \(3^{-2} = 1/3^2 = 1/9\). So the expression becomes \(log_3 (3^{-2})\).

Step 3: Apply the rule of logarithm

In the rule of logarithm, \(log_b (b^x) = x\), the base (b) is the same as the base of the power in the argument. Therefore, \(log_3 (3^{-2}) = -2\).