Precalculus Examples

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

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

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

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

Step 2.2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 \cdot 5 = x \cdot 1$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Rewrite the equation as $x \cdot 1 = 1 \cdot 5$ .

$x \cdot 1 = 1 \cdot 5$

Step 2.3.2

Multiply $x$ by $1$ .

$x = 1 \cdot 5$

Step 2.3.3

Multiply $5$ by $1$ .

$x = 5$