Question

Show that \(f=g\) by using a graphing utility to graph \(f\) and \(g\) in the same viewing window. (Assume \(x>0 .)\) $$ \begin{array}{l} f(x)=\ln \left(x^{2} / 4\right) \\ g(x)=2 \ln x-\ln 4 \end{array} $$

Short answer

The simplified version of \(f(x)\) is \(2\ln(x) - \ln(4)\), and it's seen that it's the same as \(g(x)\), so \(f=g\). This is also visually confirmed on the graph where both functions produce the same graph.

Step-by-step solution

Simplify Function \(f(x)\)

Function \(f(x)\) initially is given as \(\ln \left(x^{2} / 4\right)\). By using logarithmic properties, it can be simplified as follows: \(\ln \left(x^{2} / 4\right) = \ln(x^2) - \ln(4)\). Next, apply another logarithmic property \(\ln x^n = n \ln x\) to \(\ln(x^2)\), making it \(2\ln(x)\). Thus, \(f(x)\) simplified becomes \(2\ln(x) - \ln(4)\).

Compare \(f(x)\) and \(g(x)\)

Comparatively, the function \(g(x)\) is provided as \(2 \ln x-\ln 4\). It's clear that the simplified \(f(x)\) is identical to \(g(x)\).

Graph the Functions

For final verification, graph both functions using a graphing utility. Since they are the same, they should produce the same graph. Keep the viewing window for both graphs identical. Since \(x>0\), the graph should only show the positive x-axis.