Precalculus Examples

$\ln (3 x) - \ln (4) = 2$

Step 1

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

$\ln (\frac{3 x}{4}) = 2$

Step 2

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

$e^{\ln (\frac{3 x}{4})} = e^{2}$

Step 3

Rewrite $\ln (\frac{3 x}{4}) = 2$ 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 $b^{y} = x$ .

$e^{2} = \frac{3 x}{4}$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $\frac{3 x}{4} = e^{2}$ .

$\frac{3 x}{4} = e^{2}$

Step 4.2

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} e^{2}$

Step 4.3

Simplify both sides of the equation.

Tap for more steps...

Step 4.3.1

Simplify the left side.

Tap for more steps...

Step 4.3.1.1

Simplify $\frac{4}{3} \cdot \frac{3 x}{4}$ .

Tap for more steps...

Step 4.3.1.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.3.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} e^{2}$

Step 4.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{4}{3} e^{2}$

Step 4.3.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{4}{3} e^{2}$

Step 4.3.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{4}{3} e^{2}$

Step 4.3.1.1.2.3

Rewrite the expression.

$x = \frac{4}{3} e^{2}$

Step 4.3.2

Simplify the right side.

Tap for more steps...

Step 4.3.2.1

Combine $\frac{4}{3}$ and $e^{2}$ .

$x = \frac{4 e^{2}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 e^{2}}{3}$

Decimal Form:

$x = 9.85207479 \dots$