Evaluate log base 8 of 1/256

$\log_{8} (\frac{1}{256})$

Step 1

Rewrite as an equation.

$\log_{8} (\frac{1}{256}) = x$

Step 2

Rewrite $\log_{8} (\frac{1}{256}) = x$ 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 $b^{y} = x$ .

$8^{x} = \frac{1}{256}$

Step 3

Create expressions in the equation that all have equal bases.

${(2^{3})}^{x} = 2^{- 8}$

Step 4

Rewrite ${(2^{3})}^{x}$ as $2^{3 x}$ .

$2^{3 x} = 2^{- 8}$

Step 5

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

$3 x = - 8$

Step 6

Solve for $x$ .

$x = - \frac{8}{3}$

Step 7

The variable $x$ is equal to $- \frac{8}{3}$ .

$- \frac{8}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{3}$

Decimal Form:

$- 2.\overline{6}$

Mixed Number Form:

$- 2 \frac{2}{3}$

Evaluate log base 8 of 1/256

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