Question

The logarithmic function$f\left(x\right)=\ln \left(x-1\right)$ has the _____ asymptote $x=$_____.

Short answer

The logarithmic function $f\left(x\right)=\ln \left(x-1\right)$has the vertical asymptote $x=1$.

Step-by-step solution

Step 1. Given.

The given function is $f\left(x\right)=\ln \left(x-1\right)$.

Step 2. To determine.

We have to find the asymptote of $f\left(x\right)=\ln \left(x-1\right)$.

Step 3. Calculation.

The parent function of $f\left(x\right)=\ln \left(x-1\right)$is $g\left(x\right)=\ln \left(x\right)$.

We can write$f\left(x\right)$ using$g\left(x\right)$ as:

$$\begin{gathered} f\left(x\right)=\ln \left(x-1\right) \\ =g(x-1) \end{gathered}$$

So,$g\left(x\right)$ moves 1 unit to the right to form $f\left(x\right)$. We know that the vertical asymptote of$g\left(x\right)=\ln \left(x\right)$ is $x=0$. So, the vertical asymptote of$f\left(x\right)=\ln \left(x-1\right)$ will be $x=1$.

We can check this using a graphing calculator too:

We’ll graph the function $f\left(x\right)=\ln \left(x-1\right)$using a graphing calculator.

From this graph we see that the vertical asymptote is $x=1$.

Hence, the logarithmic function $f\left(x\right)=\ln \left(x-1\right)$has the vertical asymptote $x=1$.