select all functions whose graph has a vertical asymptote at x = 4.\na. (f(x)=log_{4}x - 4)\nb. (f(x)=ln(x…

Explanation:

Step1: Recall vertical - asymptote rule for logarithmic functions

The vertical asymptote of the logarithmic function $y = \log_a(u)$ or $y=\ln(u)$ occurs when $u = 0$.

Step2: Analyze function A

For $f(x)=\log_4x - 4$, the argument of the logarithm is $u = x$. The vertical asymptote is at $x = 0$, not $x = 4$.

Step3: Analyze function B

For $f(x)=\ln(x - 4)$, set the argument $u=x - 4=0$. Solving $x-4 = 0$ gives $x = 4$. So it has a vertical asymptote at $x = 4$.

Step4: Analyze function C

For $f(x)=\log(x - 4)+4$, set the argument $u=x - 4=0$. Solving $x - 4=0$ gives $x = 4$. So it has a vertical asymptote at $x = 4$.

Step5: Analyze function D

For $f(x)=4\ln x-4$, the argument of the logarithm is $u = x$. The vertical asymptote is at $x = 0$, not $x = 4$.

Step6: Analyze function E

For $f(x)=\log(x - 4)$, set the argument $u=x - 4=0$. Solving $x - 4=0$ gives $x = 4$. So it has a vertical asymptote at $x = 4$.

Answer:

B. $f(x)=\ln(x - 4)$ C. $f(x)=\log(x - 4)+4$ E. $f(x)=\log(x - 4)$