Precalculus Examples

$2000 = 500 e^{4 x}$

Step 1

Rewrite the equation as $500 e^{4 x} = 2000$ .

$500 e^{4 x} = 2000$

Step 2

Divide each term in $500 e^{4 x} = 2000$ by $500$ and simplify.

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Step 2.1

Divide each term in $500 e^{4 x} = 2000$ by $500$ .

$\frac{500 e^{4 x}}{500} = \frac{2000}{500}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $500$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{500 e^{4 x}}{500} = \frac{2000}{500}$

Step 2.2.1.2

Divide $e^{4 x}$ by $1$ .

$e^{4 x} = \frac{2000}{500}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $2000$ by $500$ .

$e^{4 x} = 4$

Step 3

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

$\ln (e^{4 x}) = \ln (4)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{4 x})$ by moving $4 x$ outside the logarithm.

$4 x \ln (e) = \ln (4)$

Step 4.2

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

$4 x \cdot 1 = \ln (4)$

Step 4.3

Multiply $4$ by $1$ .

$4 x = \ln (4)$

Step 5

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

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Step 5.1

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

$\frac{4 x}{4} = \frac{\ln (4)}{4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\ln (4)}{4}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (4)}{4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

Step 5.3.2

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

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

Step 5.3.3

Cancel the common factor of $2$ and $4$ .

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Step 5.3.3.1

Factor $2$ out of $2 \ln (2)$ .

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

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $2$ out of $4$ .

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

Step 5.3.3.2.2

Cancel the common factor.

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

Step 5.3.3.2.3

Rewrite the expression.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.34657359 \dots$