Algebra Examples

$\log_{- \frac{1}{2}} (x) = - 7$

Step 1

Rewrite $\log_{- \frac{1}{2}} (x) = - 7$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

${(- \frac{1}{2})}^{- 7} = x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x = {(- \frac{1}{2})}^{- 7}$ .

$x = {(- \frac{1}{2})}^{- 7}$

Step 2.2

Simplify ${(- \frac{1}{2})}^{- 7}$ .

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Step 2.2.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$x = {(- 2)}^{7}$

Step 2.2.2

Raise $- 2$ to the power of $7$ .

$x = - 128$

Step 3

Exclude the solutions that do not make $\log_{- \frac{1}{2}} (x) = - 7$ true.

$No solution$