Precalculus Examples

$\frac{1}{2} \cdot e^{3 x} = \frac{7}{2}$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot e^{3 x}) = 2 (\frac{7}{2})$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $2 (\frac{1}{2} \cdot e^{3 x})$ .

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Step 2.1.1.1

Combine $\frac{1}{2}$ and $e^{3 x}$ .

$2 \frac{e^{3 x}}{2} = 2 (\frac{7}{2})$

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Cancel the common factor.

$2 \frac{e^{3 x}}{2} = 2 (\frac{7}{2})$

Step 2.1.1.2.2

Rewrite the expression.

$e^{3 x} = 2 (\frac{7}{2})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$e^{3 x} = 2 (\frac{7}{2})$

Step 2.2.1.2

Rewrite the expression.

$e^{3 x} = 7$

Step 3

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

$\ln (e^{3 x}) = \ln (7)$

Step 4

Expand the left side.

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Step 4.1

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

$3 x \ln (e) = \ln (7)$

Step 4.2

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

$3 x \cdot 1 = \ln (7)$

Step 4.3

Multiply $3$ by $1$ .

$3 x = \ln (7)$

Step 5

Divide each term in $3 x = \ln (7)$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = \ln (7)$ by $3$ .

$\frac{3 x}{3} = \frac{\ln (7)}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\ln (7)}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (7)}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (7)}{3}$

Decimal Form:

$x = 0.64863671 \dots$