Precalculus Examples

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

Step 1

Move all terms not containing $\ln (x - 3)$ to the right side of the equation.

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Step 1.1

Subtract $2$ from both sides of the equation.

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

Step 1.2

Subtract $2$ from $3$ .

$3 \ln (x - 3) = 1$

Step 2

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

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Step 2.1

Divide each term in $3 \ln (x - 3) = 1$ by $3$ .

$\frac{3 \ln (x - 3)}{3} = \frac{1}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 \ln (x - 3)}{3} = \frac{1}{3}$

Step 2.2.1.2

Divide $\ln (x - 3)$ by $1$ .

$\ln (x - 3) = \frac{1}{3}$

Step 3

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

$e^{\ln (x - 3)} = e^{\frac{1}{3}}$

Step 4

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

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

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $x - 3 = e^{\frac{1}{3}}$ .

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

Step 5.2

Add $3$ to both sides of the equation.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 4.39561242 \dots$