Precalculus Examples

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

Step 1

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{1}{x})} = e^{- 5}$

Step 2

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

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

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{1}{x} = e^{- 5}$ .

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

Step 3.2

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

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

Step 3.3

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.

$1 e^{5} = x \cdot 1$

Step 3.4

Solve the equation for $x$ .

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Step 3.4.1

Rewrite the equation as $x \cdot 1 = 1 e^{5}$ .

$x \cdot 1 = 1 e^{5}$

Step 3.4.2

Multiply $x$ by $1$ .

$x = 1 e^{5}$

Step 3.4.3

Multiply $e^{5}$ by $1$ .

$x = e^{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = e^{5}$

Decimal Form:

$x = 148.41315910 \dots$