Precalculus Examples

$e^{3 x - 3} \cdot e^{- x} = 3 e$

Step 1

Multiply $e^{3 x - 3}$ by $e^{- x}$ by adding the exponents.

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Step 1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{3 x - 3 - x} = 3 e$

Step 1.2

Subtract $x$ from $3 x$ .

$e^{2 x - 3} = 3 e$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{2 x - 3})$ by moving $2 x - 3$ outside the logarithm.

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

Step 3.2

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

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

Step 3.3

Multiply $2 x - 3$ by $1$ .

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

Step 4

Expand the right side.

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Step 4.1

Rewrite $\ln (3 e)$ as $\ln (3) + \ln (e)$ .

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Add $3$ to both sides of the equation.

$2 x = \ln (3) + 1 + 3$

Step 5.2

Add $1$ and $3$ .

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

Step 6

Divide each term in $2 x = \ln (3) + 4$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 x = \ln (3) + 4$ by $2$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $4$ by $2$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.54930614 \dots$