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Chapter 1: Problem 73

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

Expert verified
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

01

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: \(g(x)=\ln x^2-\ln 4\)
02

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 \(g(x)=\ln (x^2/4)\)
03

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

Now that \(g(x)\) has been simplified, compare it with \(f(x)\). They are identical, hence, \(f(x)=g(x)\)
04

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.

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Most popular questions from this chapter

Prove that if \(\lim _{x \rightarrow c} f(x)\) exists and \(\lim _{x \rightarrow c}[f(x)+g(x)]\) does not exist, then \(\lim _{x \rightarrow c} g(x)\) does not exist.

True or False? In Exercises \(50-53\), determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graphs of polynomial functions have no vertical asymptotes.

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If the functions \(f\) and \(g\) are continuous for all real \(x\), is \(f+g\) always continuous for all real \(x ?\) Is \(f / g\) always continuous for all real \(x ?\) If either is not continuous, give an example to verify your conclusion.
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