Calculus Examples

$5 - \ln (x - 3) = 2$

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 $5$ from both sides of the equation.

$- \ln (x - 3) = 2 - 5$

Step 1.2

Subtract $5$ from $2$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2

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

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 3$ by $- 1$ .

$\ln (x - 3) = 3$

Step 3

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

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

Step 4

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

$e^{3} = x - 3$

Step 5

Solve for $x$ .

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Step 5.1

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

$x - 3 = e^{3}$

Step 5.2

Add $3$ to both sides of the equation.

$x = e^{3} + 3$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = e^{3} + 3$

Decimal Form:

$x = 23.08553692 \dots$