Precalculus Examples

$\log_{x} (\frac{1}{3125}) = 5$

Step 1

Rewrite $\log_{x} (\frac{1}{3125}) = 5$ 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 $b^{y} = x$ .

$x^{5} = \frac{1}{3125}$

Step 2

Solve for $x$ .

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Step 2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[5]{\frac{1}{3125}}$

Step 2.2

Simplify $\sqrt[5]{\frac{1}{3125}}$ .

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Step 2.2.1

Rewrite $\sqrt[5]{\frac{1}{3125}}$ as $\frac{\sqrt[5]{1}}{\sqrt[5]{3125}}$ .

$x = \frac{\sqrt[5]{1}}{\sqrt[5]{3125}}$

Step 2.2.2

Any root of $1$ is $1$ .

$x = \frac{1}{\sqrt[5]{3125}}$

Step 2.2.3

Simplify the denominator.

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Step 2.2.3.1

Rewrite $3125$ as $5^{5}$ .

$x = \frac{1}{\sqrt[5]{5^{5}}}$

Step 2.2.3.2

Pull terms out from under the radical, assuming real numbers.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.2$