Calculus Examples

$42 + 3 e^{\frac{3 x}{2}} = 965$

Step 1

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

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Step 1.1

Subtract $42$ from both sides of the equation.

$3 e^{\frac{3 x}{2}} = 965 - 42$

Step 1.2

Subtract $42$ from $965$ .

$3 e^{\frac{3 x}{2}} = 923$

Step 2

Divide each term in $3 e^{\frac{3 x}{2}} = 923$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 e^{\frac{3 x}{2}} = 923$ by $3$ .

$\frac{3 e^{\frac{3 x}{2}}}{3} = \frac{923}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 e^{\frac{3 x}{2}}}{3} = \frac{923}{3}$

Step 2.2.1.2

Divide $e^{\frac{3 x}{2}}$ by $1$ .

$e^{\frac{3 x}{2}} = \frac{923}{3}$

Step 3

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

$\ln (e^{\frac{3 x}{2}}) = \ln (\frac{923}{3})$

Step 4

Expand the left side.

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Step 4.1

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

$\frac{3 x}{2} \ln (e) = \ln (\frac{923}{3})$

Step 4.2

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

$\frac{3 x}{2} \cdot 1 = \ln (\frac{923}{3})$

Step 4.3

Multiply $\frac{3 x}{2}$ by $1$ .

$\frac{3 x}{2} = \ln (\frac{923}{3})$

Step 5

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \ln (\frac{923}{3})$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Step 6.1.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \ln (\frac{923}{3})$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} \ln (\frac{923}{3})$

Step 6.1.1.2

Cancel the common factor of $3$ .

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Step 6.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} \ln (\frac{923}{3})$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} \ln (\frac{923}{3})$

Step 6.1.1.2.3

Rewrite the expression.

$x = \frac{2}{3} \ln (\frac{923}{3})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Combine $\frac{2}{3}$ and $\ln (\frac{923}{3})$ .

$x = \frac{2 \ln (\frac{923}{3})}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \ln (\frac{923}{3})}{3}$

Decimal Form:

$x = 3.81934463 \dots$