Precalculus Examples

$2 e^{5 a + 2} = 8$

Step 1

Divide each term in $2 e^{5 a + 2} = 8$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 e^{5 a + 2} = 8$ by $2$ .

$\frac{2 e^{5 a + 2}}{2} = \frac{8}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 e^{5 a + 2}}{2} = \frac{8}{2}$

Step 1.2.1.2

Divide $e^{5 a + 2}$ by $1$ .

$e^{5 a + 2} = \frac{8}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $8$ by $2$ .

$e^{5 a + 2} = 4$

Step 2

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

$\ln (e^{5 a + 2}) = \ln (4)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{5 a + 2})$ by moving $5 a + 2$ outside the logarithm.

$(5 a + 2) \ln (e) = \ln (4)$

Step 3.2

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

$(5 a + 2) \cdot 1 = \ln (4)$

Step 3.3

Multiply $5 a + 2$ by $1$ .

$5 a + 2 = \ln (4)$

Step 4

Subtract $2$ from both sides of the equation.

$5 a = \ln (4) - 2$

Step 5

Divide each term in $5 a = \ln (4) - 2$ by $5$ and simplify.

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Step 5.1

Divide each term in $5 a = \ln (4) - 2$ by $5$ .

$\frac{5 a}{5} = \frac{\ln (4)}{5} + \frac{- 2}{5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $5$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{5 a}{5} = \frac{\ln (4)}{5} + \frac{- 2}{5}$

Step 5.2.1.2

Divide $a$ by $1$ .

$a = \frac{\ln (4)}{5} + \frac{- 2}{5}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$a = \frac{\ln (4)}{5} - \frac{2}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{\ln (4)}{5} - \frac{2}{5}$

Decimal Form:

$a = - 0.12274112 \dots$