Problem 28 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

-2

Step 1: Understanding the Logarithm

In a logarithmic expression \(\log_b a = n\), 'b' is the base, 'a' is the number and 'n' is the exponent to which the base must be raised to obtain the number. So, \(\log _{b} a = n\) is equivalent to \(b^n = a\).

Step 2: Reshape the Number

Given the number as 1/9, it can be written as \(3^{-2}\) since \(3^{-2} = 1/(3^2) = 1/9\).

Step 3: Evaluate the Expression

Substitute \(3^{-2}\) in place of 1/9 in the original expression, so we get \(\log_3{(3^{-2})}\). Now according to the logarithm rule, the expression equals to -2.