Evaluate log base 1/3 of 1/9

$\log_{\frac{1}{3}} (\frac{1}{9})$

Step 1

Rewrite as an equation.

$\log_{\frac{1}{3}} (\frac{1}{9}) = x$

Step 2

Rewrite $\log_{\frac{1}{3}} (\frac{1}{9}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

${(\frac{1}{3})}^{x} = \frac{1}{9}$

Step 3

Create expressions in the equation that all have equal bases.

${(3^{- 1})}^{x} = 3^{- 2}$

Step 4

Rewrite ${(3^{- 1})}^{x}$ as $3^{- x}$ .

$3^{- x} = 3^{- 2}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$- x = - 2$

Step 6

Solve for $x$ .

$x = 2$

Step 7

The variable $x$ is equal to $2$ .

$2$

Evaluate log base 1/3 of 1/9

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