Precalculus Examples

$\log_{5} (x) = - 3$

Step 1

Rewrite $\log_{5} (x) = - 3$ 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 $b^{y} = x$ .

$5^{- 3} = x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x = 5^{- 3}$ .

$x = 5^{- 3}$

Step 2.2

Simplify $5^{- 3}$ .

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Step 2.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{5^{3}}$

Step 2.2.2

Raise $5$ to the power of $3$ .

$x = \frac{1}{125}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{125}$

Decimal Form:

$x = 0.008$