Precalculus Examples

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

Step 1

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

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

Step 2

Expand the left side.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $x + 3$ by $1$ .

$x + 3 = \ln (3^{x})$

Step 3

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

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

Step 4

Subtract $x \ln (3)$ from both sides of the equation.

$x + 3 - x \ln (3) = 0$

Step 5

Subtract $3$ from both sides of the equation.

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

Step 6

Factor the left side of the equation.

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Step 6.1

Factor $x$ out of $x - x \ln (3)$ .

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Step 6.1.1

Raise $x$ to the power of $1$ .

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

Step 6.1.2

Factor $x$ out of $x^{1}$ .

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

Step 6.1.3

Factor $x$ out of $- x \ln (3)$ .

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

Step 6.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 \ln (3))$ .

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

Step 6.2

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

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

Step 7

Divide each term in $x (1 - \ln (3)) = - 3$ by $1 - \ln (3)$ and simplify.

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Step 7.1

Divide each term in $x (1 - \ln (3)) = - 3$ by $1 - \ln (3)$ .

$\frac{x (1 - \ln (3))}{1 - \ln (3)} = \frac{- 3}{1 - \ln (3)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $1 - \ln (3)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{x (1 - \ln (3))}{1 - \ln (3)} = \frac{- 3}{1 - \ln (3)}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{1 - \ln (3)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{1 - \ln (3)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{1 - \ln (3)}$

Decimal Form:

$x = 30.42217192 \dots$