Precalculus Examples

$\ln (x - 1) = 2$

Step 1

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

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

Step 2

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

$e^{2} = x - 1$

Step 3

Solve for $x$ .

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Step 3.1

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

$x - 1 = e^{2}$

Step 3.2

Add $1$ to both sides of the equation.

$x = e^{2} + 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = e^{2} + 1$

Decimal Form:

$x = 8.38905609 \dots$