Question

use the change-of-base formula $\log_{a}(x) = \frac{\ln(x)}{\ln(a)}$ and a graphing utility to graph the function. $f(x) = \log_{1/5}(x + 1)$

Explanation:

Step1: Apply Change - of - Base Formula

Given the function \(f(x)=\log_{\frac{1}{5}}(x + 1)\), using the change - of - base formula \(\log_{a}(x)=\frac{\ln(x)}{\ln(a)}\), we substitute \(a = \frac{1}{5}\) and \(x=(x + 1)\). So we get \(f(x)=\frac{\ln(x + 1)}{\ln(\frac{1}{5})}\).

Step2: Analyze the Domain and Asymptote

The argument of the logarithm \(x + 1>0\), so the domain is \(x>- 1\). The vertical asymptote is \(x=-1\) (since as \(x
ightarrow - 1^{+}\), \(\ln(x + 1)
ightarrow-\infty\) and \(\ln(\frac{1}{5})\) is a constant (\(\ln(\frac{1}{5})=-\ln(5)<0\))).

Step3: Analyze the Behavior of the Function

The base of the logarithm \(a=\frac{1}{5}<1\), so the function \(y = \log_{\frac{1}{5}}u\) is a decreasing function of \(u\). Here \(u=x + 1\) is an increasing function of \(x\). By the composition of functions (a decreasing function of an increasing function is a decreasing function), \(f(x)=\log_{\frac{1}{5}}(x + 1)\) is a decreasing function.

Step4: Identify the Correct Graph

• For the left graph: The vertical asymptote seems to be at \(x = 0\) (incorrect, our asymptote is \(x=-1\)) and the function passes through \((0,0)\). Let's check \(f(0)=\log_{\frac{1}{5}}(0 + 1)=\log_{\frac{1}{5}}(1) = 0\), but the asymptote is wrong.
• For the right graph: The vertical asymptote is at \(x=-1\) (correct, since the domain is \(x>-1\)). When \(x = 0\), \(f(0)=\log_{\frac{1}{5}}(1)=0\)? Wait, no, wait: \(f(0)=\log_{\frac{1}{5}}(0 + 1)=\log_{\frac{1}{5}}(1) = 0\)? Wait, no, \(\log_{a}(1)=0\) for any \(a>0,a

eq1\). Wait, when \(x = 0\), \(y = 0\)? Wait, no, let's recalculate \(f(0)=\frac{\ln(1)}{\ln(\frac{1}{5})}=\frac{0}{\ln(\frac{1}{5})}=0\). But when \(x = 0\), the right graph has \(y\) - value around \(0\)? Wait, no, let's check the left - hand side of the vertical asymptote. The left graph has the asymptote at \(x = 0\) (wrong), the right graph has the asymptote at \(x=-1\) (correct). Also, as \(x\) increases (moves to the right from \(x=-1\)), since the function is decreasing, the \(y\) - value should decrease. The right graph shows a decreasing function with vertical asymptote \(x = - 1\) (the left - most part of the right graph is near \(x=-1\) and as \(x\) increases, \(y\) decreases), which matches our function. The left graph has an asymptote at \(x = 0\) and is decreasing but with the wrong asymptote. So the correct graph is the right graph.

Answer:

The Right Graph