Precalculus Examples

$2 \ln (3 x + 8) = 28$

Step 1

Divide each term in $2 \ln (3 x + 8) = 28$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \ln (3 x + 8) = 28$ by $2$ .

$\frac{2 \ln (3 x + 8)}{2} = \frac{28}{2}$

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.

$\frac{2 \ln (3 x + 8)}{2} = \frac{28}{2}$

Step 1.2.1.2

Divide $\ln (3 x + 8)$ by $1$ .

$\ln (3 x + 8) = \frac{28}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $28$ by $2$ .

$\ln (3 x + 8) = 14$

Step 2

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (3 x + 8)} = e^{14}$

Step 3

Rewrite $\ln (3 x + 8) = 14$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{14} = 3 x + 8$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $3 x + 8 = e^{14}$ .

$3 x + 8 = e^{14}$

Step 4.2

Subtract $8$ from both sides of the equation.

$3 x = e^{14} - 8$

Step 4.3

Divide each term in $3 x = e^{14} - 8$ by $3$ and simplify.

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Step 4.3.1

Divide each term in $3 x = e^{14} - 8$ by $3$ .

$\frac{3 x}{3} = \frac{e^{14}}{3} + \frac{- 8}{3}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $3$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{e^{14}}{3} + \frac{- 8}{3}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{14}}{3} + \frac{- 8}{3}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Move the negative in front of the fraction.

$x = \frac{e^{14}}{3} - \frac{8}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{14}}{3} - \frac{8}{3}$

Decimal Form:

$x = 400865.42805492 \dots$