Precalculus Examples

$\ln (x) - \ln (3) = 2$

Step 1

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

$\ln (\frac{x}{3}) = 2$

Step 2

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

$e^{\ln (\frac{x}{3})} = e^{2}$

Step 3

Rewrite $\ln (\frac{x}{3}) = 2$ 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 $b^{y} = x$ .

$e^{2} = \frac{x}{3}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{x}{3} = e^{2}$ .

$\frac{x}{3} = e^{2}$

Step 4.2

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 e^{2}$

Step 4.3

Simplify the left side.

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Step 4.3.1

Cancel the common factor of $3$ .

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Step 4.3.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 e^{2}$

Step 4.3.1.2

Rewrite the expression.

$x = 3 e^{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 3 e^{2}$

Decimal Form:

$x = 22.16716829 \dots$