Precalculus Examples

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

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Rewrite $\frac{e^{3 x}}{e^{2}}$ as $e^{3 x} {(e^{2})}^{- 1}$ .

$\ln (e^{3 x} {(e^{2})}^{- 1}) = 2$

Step 1.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (e^{3 x} e^{2 \cdot - 1}) = 2$

Step 1.4

Rewrite $\ln (e^{3 x} e^{2 \cdot - 1})$ as $\ln (e^{3 x}) + \ln (e^{2 \cdot - 1})$ .

$\ln (e^{3 x}) + \ln (e^{2 \cdot - 1}) = 2$

Step 1.5

Use logarithm rules to move $3 x$ out of the exponent.

$3 x \ln (e) + \ln (e^{2 \cdot - 1}) = 2$

Step 1.6

The natural logarithm of $e$ is $1$ .

$3 x \cdot 1 + \ln (e^{2 \cdot - 1}) = 2$

Step 1.7

Multiply $3$ by $1$ .

$3 x + \ln (e^{2 \cdot - 1}) = 2$

Step 1.8

Simplify each term.

Tap for more steps...

Step 1.8.1

Use logarithm rules to move $2 \cdot - 1$ out of the exponent.

$3 x + 2 \cdot (- 1 \ln (e)) = 2$

Step 1.8.2

Multiply $2$ by $- 1$ .

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

Step 1.8.3

The natural logarithm of $e$ is $1$ .

$3 x - 2 \cdot 1 = 2$

Step 1.8.4

Multiply $- 2$ by $1$ .

$3 x - 2 = 2$

Step 2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 2.1

Add $2$ to both sides of the equation.

$3 x = 2 + 2$

Step 2.2

Add $2$ and $2$ .

$3 x = 4$

Step 3

Divide each term in $3 x = 4$ by $3$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{3}$

Decimal Form:

$x = 1.\overline{3}$

Mixed Number Form:

$x = 1 \frac{1}{3}$