Calculus Examples

$e^{3 - 4 x} = 4$

Step 1

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

$\ln (e^{3 - 4 x}) = \ln (4)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{3 - 4 x})$ by moving $3 - 4 x$ outside the logarithm.

$(3 - 4 x) \ln (e) = \ln (4)$

Step 2.2

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

$(3 - 4 x) \cdot 1 = \ln (4)$

Step 2.3

Multiply $3 - 4 x$ by $1$ .

$3 - 4 x = \ln (4)$

Step 3

Subtract $3$ from both sides of the equation.

$- 4 x = \ln (4) - 3$

Step 4

Divide each term in $- 4 x = \ln (4) - 3$ by $- 4$ and simplify.

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Step 4.1

Divide each term in $- 4 x = \ln (4) - 3$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{\ln (4)}{- 4} + \frac{- 3}{- 4}$

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.

$\frac{- 4 x}{- 4} = \frac{\ln (4)}{- 4} + \frac{- 3}{- 4}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (4)}{- 4} + \frac{- 3}{- 4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

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

$x = \frac{\ln (2^{2})}{- 4} + \frac{- 3}{- 4}$

Step 4.3.1.2

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

$x = \frac{2 \ln (2)}{- 4} + \frac{- 3}{- 4}$

Step 4.3.1.3

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

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Step 4.3.1.3.1

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

$x = \frac{2 (\ln (2))}{- 4} + \frac{- 3}{- 4}$

Step 4.3.1.3.2

Cancel the common factors.

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Step 4.3.1.3.2.1

Factor $2$ out of $- 4$ .

$x = \frac{2 \ln (2)}{2 \cdot - 2} + \frac{- 3}{- 4}$

Step 4.3.1.3.2.2

Cancel the common factor.

$x = \frac{2 \ln (2)}{2 \cdot - 2} + \frac{- 3}{- 4}$

Step 4.3.1.3.2.3

Rewrite the expression.

$x = \frac{\ln (2)}{- 2} + \frac{- 3}{- 4}$

Step 4.3.1.4

Move the negative in front of the fraction.

$x = - \frac{\ln (2)}{2} + \frac{- 3}{- 4}$

Step 4.3.1.5

Dividing two negative values results in a positive value.

$x = - \frac{\ln (2)}{2} + \frac{3}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (2)}{2} + \frac{3}{4}$

Decimal Form:

$x = 0.40342640 \dots$