Precalculus Examples

$\frac{1025}{8 + e^{4 x}} = 5$

Step 1

Multiply both sides by $8 + e^{4 x}$ .

$\frac{1025}{8 + e^{4 x}} (8 + e^{4 x}) = 5 (8 + e^{4 x})$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $8 + e^{4 x}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{1025}{8 + e^{4 x}} (8 + e^{4 x}) = 5 (8 + e^{4 x})$

Step 2.1.1.2

Rewrite the expression.

$1025 = 5 (8 + e^{4 x})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $5 (8 + e^{4 x})$ .

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Step 2.2.1.1

Apply the distributive property.

$1025 = 5 \cdot 8 + 5 e^{4 x}$

Step 2.2.1.2

Multiply $5$ by $8$ .

$1025 = 40 + 5 e^{4 x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $40 + 5 e^{4 x} = 1025$ .

$40 + 5 e^{4 x} = 1025$

Step 3.2

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

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Step 3.2.1

Subtract $40$ from both sides of the equation.

$5 e^{4 x} = 1025 - 40$

Step 3.2.2

Subtract $40$ from $1025$ .

$5 e^{4 x} = 985$

Step 3.3

Divide each term in $5 e^{4 x} = 985$ by $5$ and simplify.

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Step 3.3.1

Divide each term in $5 e^{4 x} = 985$ by $5$ .

$\frac{5 e^{4 x}}{5} = \frac{985}{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{5 e^{4 x}}{5} = \frac{985}{5}$

Step 3.3.2.1.2

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

$e^{4 x} = \frac{985}{5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $985$ by $5$ .

$e^{4 x} = 197$

Step 3.4

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

$\ln (e^{4 x}) = \ln (197)$

Step 3.5

Expand the left side.

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Step 3.5.1

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

$4 x \ln (e) = \ln (197)$

Step 3.5.2

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

$4 x \cdot 1 = \ln (197)$

Step 3.5.3

Multiply $4$ by $1$ .

$4 x = \ln (197)$

Step 3.6

Divide each term in $4 x = \ln (197)$ by $4$ and simplify.

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Step 3.6.1

Divide each term in $4 x = \ln (197)$ by $4$ .

$\frac{4 x}{4} = \frac{\ln (197)}{4}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $4$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\ln (197)}{4}$

Step 3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (197)}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (197)}{4}$

Decimal Form:

$x = 1.32080093 \dots$