Precalculus Examples

$3 \ln (2 x + 1) = 9$

Step 1

Divide each term in $3 \ln (2 x + 1) = 9$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 \ln (2 x + 1) = 9$ by $3$ .

$\frac{3 \ln (2 x + 1)}{3} = \frac{9}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 \ln (2 x + 1)}{3} = \frac{9}{3}$

Step 1.2.1.2

Divide $\ln (2 x + 1)$ by $1$ .

$\ln (2 x + 1) = \frac{9}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $9$ by $3$ .

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

Step 2

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

$e^{\ln (2 x + 1)} = e^{3}$

Step 3

Rewrite $\ln (2 x + 1) = 3$ 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 $b^{y} = x$ .

$e^{3} = 2 x + 1$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $2 x + 1 = e^{3}$ .

$2 x + 1 = e^{3}$

Step 4.2

Subtract $1$ from both sides of the equation.

$2 x = e^{3} - 1$

Step 4.3

Divide each term in $2 x = e^{3} - 1$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 x = e^{3} - 1$ by $2$ .

$\frac{2 x}{2} = \frac{e^{3}}{2} + \frac{- 1}{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{e^{3}}{2} + \frac{- 1}{2}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{3}}{2} + \frac{- 1}{2}$

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.

$x = \frac{e^{3}}{2} - \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{3}}{2} - \frac{1}{2}$

Decimal Form:

$x = 9.54276846 \dots$