Precalculus Examples

$e^{4 x + 1} = 200$

Step 1

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

$\ln (e^{4 x + 1}) = \ln (200)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{4 x + 1})$ by moving $4 x + 1$ outside the logarithm.

$(4 x + 1) \ln (e) = \ln (200)$

Step 2.2

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

$(4 x + 1) \cdot 1 = \ln (200)$

Step 2.3

Multiply $4 x + 1$ by $1$ .

$4 x + 1 = \ln (200)$

Step 3

Subtract $1$ from both sides of the equation.

$4 x = \ln (200) - 1$

Step 4

Divide each term in $4 x = \ln (200) - 1$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 x = \ln (200) - 1$ by $4$ .

$\frac{4 x}{4} = \frac{\ln (200)}{4} + \frac{- 1}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\ln (200)}{4} + \frac{- 1}{4}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (200)}{4} + \frac{- 1}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = \frac{\ln (200)}{4} - \frac{1}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (200)}{4} - \frac{1}{4}$

Decimal Form:

$x = 1.07457934 \dots$