Question

what is the value of \\(\log_{6} \frac{1}{36}\\)?

\\(\bigcirc\\) \\(-6\\)
\\(\bigcirc\\) \\(-2\\)
\\(\bigcirc\\) \\(2\\)
\\(\bigcirc\\) \\(6\\)

Explanation:

⚡ Using what you learned: Evaluating Logarithmic Expressions

Step 1: Set up the equation

Set the expression equal to \( x \):

$$ \log_{6} \frac{1}{36} = x $$

Step 2: Rewrite in exponential form

Convert the logarithmic equation to its exponential form \( b^x = y \):

$$ 6^x = \frac{1}{36} $$

Step 3: Express with a common base

Write \( \frac{1}{36} \) as a power of \( 6 \):

$$ \frac{1}{36} = \frac{1}{6^2} = 6^{-2} $$

Thus:

$$ 6^x = 6^{-2} $$

$$ x = -2 $$

Answer:

-2