Question

evaluate the expression without using a calculator.
\log_{9} \frac{1}{9}

Explanation:

Step1: Recall logarithm definition

Let \( y = \log_{9}\frac{1}{9} \). By definition of logarithm, \( 9^{y}=\frac{1}{9} \).

Step2: Express as same base

Note that \( \frac{1}{9}=9^{-1} \), so \( 9^{y}=9^{-1} \).

Step3: Equate exponents

Since bases are equal, exponents must be equal. Thus, \( y = - 1 \).

Answer:

\(-1\)