Precalculus Examples

$\ln (3 x + 5) = 8$

Step 1

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

$e^{\ln (3 x + 5)} = e^{8}$

Step 2

Rewrite $\ln (3 x + 5) = 8$ 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 $b^{y} = x$ .

$e^{8} = 3 x + 5$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $3 x + 5 = e^{8}$ .

$3 x + 5 = e^{8}$

Step 3.2

Subtract $5$ from both sides of the equation.

$3 x = e^{8} - 5$

Step 3.3

Divide each term in $3 x = e^{8} - 5$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 x = e^{8} - 5$ by $3$ .

$\frac{3 x}{3} = \frac{e^{8}}{3} + \frac{- 5}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{e^{8}}{3} + \frac{- 5}{3}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{8}}{3} + \frac{- 5}{3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$x = \frac{e^{8}}{3} - \frac{5}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{8}}{3} - \frac{5}{3}$

Decimal Form:

$x = 991.98599568 \dots$