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 functions \(f(x)\) and \(g(x)\) are equal. This is demonstrated algebraically by simplifying \(g(x)\) using log properties and by visually through graphing.

Step-by-step solution

Simplify \(g(x)\) using log properties

The properties of logarithms allows the equation \(g(x)=2 \ln x-\ln 4\) to be simplified. This can be done by using the rule \(\log b(a^c) = c\log b(a)\). So, the equation can be expressed as follows: \(g(x)=\ln x^2-\ln 4\)

Apply the law of logarithms

The law of logarithms can be used to further simplify \(g(x)\). The law \(\log b(a) - \log b(c) = \log b(a/c)\) can be applied to get \(g(x)=\ln (x^2/4)\)

Comparison of \(f(x)\) and \(g(x)\)

Now that \(g(x)\) has been simplified, compare it with \(f(x)\). They are identical, hence, \(f(x)=g(x)\)

Visual confirmation using graphing utility

For further verification, both functions can be plotted in a graphing utility. Since both \(f(x)\) and \(g(x)\) represent the same equation, they will produce identical graphs.