Evaluate log base 1/4 of 1/32

$\log_{\frac{1}{4}} (\frac{1}{32})$

Step 1

Rewrite as an equation.

$\log_{\frac{1}{4}} (\frac{1}{32}) = x$

Step 2

Rewrite $\log_{\frac{1}{4}} (\frac{1}{32}) = x$ 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 $b^{y} = x$ .

${(\frac{1}{4})}^{x} = \frac{1}{32}$

Step 3

Create expressions in the equation that all have equal bases.

${(2^{- 2})}^{x} = 2^{- 5}$

Step 4

Rewrite ${(2^{- 2})}^{x}$ as $2^{- 2 x}$ .

$2^{- 2 x} = 2^{- 5}$

Step 5

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

$- 2 x = - 5$

Step 6

Solve for $x$ .

$x = \frac{5}{2}$

Step 7

The variable $x$ is equal to $\frac{5}{2}$ .

$\frac{5}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2}$

Decimal Form:

$2.5$

Mixed Number Form:

$2 \frac{1}{2}$

Evaluate log base 1/4 of 1/32

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