Calculus Examples

$e^{7 x + 8} - 5 = 0$

Step 1

Add $5$ to both sides of the equation.

$e^{7 x + 8} = 5$

Step 2

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

$\ln (e^{7 x + 8}) = \ln (5)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{7 x + 8})$ by moving $7 x + 8$ outside the logarithm.

$(7 x + 8) \ln (e) = \ln (5)$

Step 3.2

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

$(7 x + 8) \cdot 1 = \ln (5)$

Step 3.3

Multiply $7 x + 8$ by $1$ .

$7 x + 8 = \ln (5)$

Step 4

Subtract $8$ from both sides of the equation.

$7 x = \ln (5) - 8$

Step 5

Divide each term in $7 x = \ln (5) - 8$ by $7$ and simplify.

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Step 5.1

Divide each term in $7 x = \ln (5) - 8$ by $7$ .

$\frac{7 x}{7} = \frac{\ln (5)}{7} + \frac{- 8}{7}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $7$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{\ln (5)}{7} + \frac{- 8}{7}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (5)}{7} + \frac{- 8}{7}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x = \frac{\ln (5)}{7} - \frac{8}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (5)}{7} - \frac{8}{7}$

Decimal Form:

$x = - 0.91293744 \dots$