Finite Math Examples

$e^{x - 4} (e^{2 x + 8}) = 24$

Step 1

Multiply $e^{x - 4}$ by $e^{2 x + 8}$ by adding the exponents.

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Step 1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{x - 4 + 2 x + 8} = 24$

Step 1.2

Add $x$ and $2 x$ .

$e^{3 x - 4 + 8} = 24$

Step 1.3

Add $- 4$ and $8$ .

$e^{3 x + 4} = 24$

Step 2

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

$\ln (e^{3 x + 4}) = \ln (24)$

Step 3

Expand the left side.

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Step 3.1

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

$(3 x + 4) \ln (e) = \ln (24)$

Step 3.2

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

$(3 x + 4) \cdot 1 = \ln (24)$

Step 3.3

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

$3 x + 4 = \ln (24)$

Step 4

Subtract $4$ from both sides of the equation.

$3 x = \ln (24) - 4$

Step 5

Divide each term in $3 x = \ln (24) - 4$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = \ln (24) - 4$ by $3$ .

$\frac{3 x}{3} = \frac{\ln (24)}{3} + \frac{- 4}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\ln (24)}{3} + \frac{- 4}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (24)}{3} + \frac{- 4}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x = \frac{\ln (24)}{3} - \frac{4}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (24)}{3} - \frac{4}{3}$

Decimal Form:

$x = - 0.27398205 \dots$