Precalculus Examples

$\ln (x) - \ln (5) = 4$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{x}{5}) = 4$

Step 2

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

$e^{\ln (\frac{x}{5})} = e^{4}$

Step 3

Rewrite $\ln (\frac{x}{5}) = 4$ 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 $b^{y} = x$ .

$e^{4} = \frac{x}{5}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{x}{5} = e^{4}$ .

$\frac{x}{5} = e^{4}$

Step 4.2

Multiply both sides of the equation by $5$ .

$5 \frac{x}{5} = 5 e^{4}$

Step 4.3

Simplify the left side.

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Step 4.3.1

Cancel the common factor of $5$ .

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Step 4.3.1.1

Cancel the common factor.

$5 \frac{x}{5} = 5 e^{4}$

Step 4.3.1.2

Rewrite the expression.

$x = 5 e^{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 5 e^{4}$

Decimal Form:

$x = 272.99075016 \dots$