Calculus Examples

$7 = - 5 + 4 e^{- 6 x}$

Step 1

Rewrite the equation as $- 5 + 4 e^{- 6 x} = 7$ .

$- 5 + 4 e^{- 6 x} = 7$

Step 2

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

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Step 2.1

Add $5$ to both sides of the equation.

$4 e^{- 6 x} = 7 + 5$

Step 2.2

Add $7$ and $5$ .

$4 e^{- 6 x} = 12$

Step 3

Divide each term in $4 e^{- 6 x} = 12$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 e^{- 6 x} = 12$ by $4$ .

$\frac{4 e^{- 6 x}}{4} = \frac{12}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 e^{- 6 x}}{4} = \frac{12}{4}$

Step 3.2.1.2

Divide $e^{- 6 x}$ by $1$ .

$e^{- 6 x} = \frac{12}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $12$ by $4$ .

$e^{- 6 x} = 3$

Step 4

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

$\ln (e^{- 6 x}) = \ln (3)$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{- 6 x})$ by moving $- 6 x$ outside the logarithm.

$- 6 x \ln (e) = \ln (3)$

Step 5.2

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

$- 6 x \cdot 1 = \ln (3)$

Step 5.3

Multiply $- 6$ by $1$ .

$- 6 x = \ln (3)$

Step 6

Divide each term in $- 6 x = \ln (3)$ by $- 6$ and simplify.

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Step 6.1

Divide each term in $- 6 x = \ln (3)$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{\ln (3)}{- 6}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 6$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{\ln (3)}{- 6}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (3)}{- 6}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (3)}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (3)}{6}$

Decimal Form:

$x = - 0.18310204 \dots$