Precalculus Examples

$14 e^{3 x + 2} = 560$

Step 1

Divide each term in $14 e^{3 x + 2} = 560$ by $14$ and simplify.

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Step 1.1

Divide each term in $14 e^{3 x + 2} = 560$ by $14$ .

$\frac{14 e^{3 x + 2}}{14} = \frac{560}{14}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $14$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{14 e^{3 x + 2}}{14} = \frac{560}{14}$

Step 1.2.1.2

Divide $e^{3 x + 2}$ by $1$ .

$e^{3 x + 2} = \frac{560}{14}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $560$ by $14$ .

$e^{3 x + 2} = 40$

Step 2

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

$\ln (e^{3 x + 2}) = \ln (40)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{3 x + 2})$ by moving $3 x + 2$ outside the logarithm.

$(3 x + 2) \ln (e) = \ln (40)$

Step 3.2

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

$(3 x + 2) \cdot 1 = \ln (40)$

Step 3.3

Multiply $3 x + 2$ by $1$ .

$3 x + 2 = \ln (40)$

Step 4

Subtract $2$ from both sides of the equation.

$3 x = \ln (40) - 2$

Step 5

Divide each term in $3 x = \ln (40) - 2$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = \ln (40) - 2$ by $3$ .

$\frac{3 x}{3} = \frac{\ln (40)}{3} + \frac{- 2}{3}$

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.

$\frac{3 x}{3} = \frac{\ln (40)}{3} + \frac{- 2}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (40)}{3} + \frac{- 2}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x = \frac{\ln (40)}{3} - \frac{2}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (40)}{3} - \frac{2}{3}$

Decimal Form:

$x = 0.56295981 \dots$