Algebra Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \ln (x - 1) - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\ln (x - 1) - 1 = 0$ .

$\ln (x - 1) - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$\ln (x - 1) = 1$

Step 1.2.3

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

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

Step 1.2.4

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

$e^{1} = x - 1$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

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

$x - 1 = e$

Step 1.2.5.2

Add $1$ to both sides of the equation.

$x = e + 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(e + 1, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \ln ((0) - 1) - 1$

Step 2.2

Solve the equation.

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Step 2.2.1

The natural logarithm of a negative number is undefined.

$y = Undefined$

Step 2.2.2

Remove parentheses.

$y = \ln ((0) - 1) - 1$

Step 2.2.3

The equation cannot be solved because it is undefined.

$Undefined$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(e + 1, 0)$

y-intercept(s): $None$

Step 4