Precalculus Examples

$6 \ln (x + 2) - 3 = 21$

Step 1

Move all terms not containing $\ln (x + 2)$ to the right side of the equation.

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Step 1.1

Add $3$ to both sides of the equation.

$6 \ln (x + 2) = 21 + 3$

Step 1.2

Add $21$ and $3$ .

$6 \ln (x + 2) = 24$

Step 2

Divide each term in $6 \ln (x + 2) = 24$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 \ln (x + 2) = 24$ by $6$ .

$\frac{6 \ln (x + 2)}{6} = \frac{24}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{6 \ln (x + 2)}{6} = \frac{24}{6}$

Step 2.2.1.2

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

$\ln (x + 2) = \frac{24}{6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $24$ by $6$ .

$\ln (x + 2) = 4$

Step 3

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

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

Step 4

Rewrite $\ln (x + 2) = 4$ 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 $b^{y} = x$ .

$e^{4} = x + 2$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $x + 2 = e^{4}$ .

$x + 2 = e^{4}$

Step 5.2

Subtract $2$ from both sides of the equation.

$x = e^{4} - 2$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = e^{4} - 2$

Decimal Form:

$x = 52.59815003 \dots$