Calculus Examples

$4 e^{4 x} - e^{- x} = 0$

Step 1

Move $- e^{- x}$ to the right side of the equation by adding it to both sides.

$4 e^{4 x} = e^{- x}$

Step 2

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

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

Step 3

Expand the left side.

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Step 3.1

Rewrite $\ln (4 e^{4 x})$ as $\ln (4) + \ln (e^{4 x})$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $4$ by $1$ .

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

Step 4

Expand the right side.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $- 1$ by $1$ .

$\ln (4) + 4 x = - x$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Add $x$ to both sides of the equation.

$\ln (4) + 4 x + x = 0$

Step 5.2

Add $4 x$ and $x$ .

$\ln (4) + 5 x = 0$

Step 6

Subtract $\ln (4)$ from both sides of the equation.

$5 x = - \ln (4)$

Step 7

Divide each term in $5 x = - \ln (4)$ by $5$ and simplify.

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Step 7.1

Divide each term in $5 x = - \ln (4)$ by $5$ .

$\frac{5 x}{5} = \frac{- \ln (4)}{5}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $5$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- \ln (4)}{5}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (4)}{5}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (4)}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (4)}{5}$

Decimal Form:

$x = - 0.27725887 \dots$