Precalculus Examples

$5 + \ln (x - 4) = 3$

Step 1

Move all terms not containing $\ln (x - 4)$ to the right side of the equation.

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Step 1.1

Subtract $5$ from both sides of the equation.

$\ln (x - 4) = 3 - 5$

Step 1.2

Subtract $5$ from $3$ .

$\ln (x - 4) = - 2$

Step 2

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

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

Step 3

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

$e^{- 2} = x - 4$

Step 4

Solve for $x$ .

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Step 4.1

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

$x - 4 = e^{- 2}$

Step 4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x - 4 = \frac{1}{e^{2}}$

Step 4.3

Add $4$ to both sides of the equation.

$x = \frac{1}{e^{2}} + 4$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{e^{2}} + 4$

Decimal Form:

$x = 4.13533528 \dots$