Precalculus Examples

$4 e^{2 x} = 40$

Step 1

Divide each term in $4 e^{2 x} = 40$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 e^{2 x} = 40$ by $4$ .

$\frac{4 e^{2 x}}{4} = \frac{40}{4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{4 e^{2 x}}{4} = \frac{40}{4}$

Step 1.2.1.2

Divide $e^{2 x}$ by $1$ .

$e^{2 x} = \frac{40}{4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $40$ by $4$ .

$e^{2 x} = 10$

Step 2

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

$\ln (e^{2 x}) = \ln (10)$

Step 3

Expand the left side.

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Step 3.1

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

$2 x \ln (e) = \ln (10)$

Step 3.2

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

$2 x \cdot 1 = \ln (10)$

Step 3.3

Multiply $2$ by $1$ .

$2 x = \ln (10)$

Step 4

Divide each term in $2 x = \ln (10)$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = \ln (10)$ by $2$ .

$\frac{2 x}{2} = \frac{\ln (10)}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\ln (10)}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (10)}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (10)}{2}$

Decimal Form:

$x = 1.15129254 \dots$