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Magazine Problem 94
Use a graphing utility to verify that the functions are equivalent for \(x>0 .\) $$ \begin{array}{l}{f(x)=\ln \sqrt{x\left(x^{2}+1\right)}} \\\ {g(x)=\frac{1}{2}\left[\ln x+\ln \left(x^{2}+1\right)\right]}\end{array} $$
Problem 95
True or False? Determine whether the statement is true or false given that \(f(x)=\ln x\). If it is false, explain why or give an example that shows it is false. $$ f(0)=0 $$
Problem 96
True or False? Determine whether the statement is true or false given that \(f(x)=\ln x\). If it is false, explain why or give an example that shows it is false. $$ f(a x)=f(a)+f(x), \quad a>0, x>0 $$
Problem 97
True or False? Determine whether the statement is true or false given that \(f(x)=\ln x\). If it is false, explain why or give an example that shows it is false. $$ f(x-2)=f(x)-f(2), \quad x>2 $$
Problem 98
True or False? Determine whether the statement is true or false given that \(f(x)=\ln x\). If it is false, explain why or give an example that shows it is false. $$ \sqrt{f(x)}=\frac{1}{2} f(x) $$
Problem 100
True or False? Determine whether the statement is true or false given that
\(f(x)=\ln x\). If it is false, explain why or give an example that shows it is
false.$$
\text { If } f(x)<0, \text { then } 0
Problem 101
Use a graphing utility to graph $$ y=10 \ln \left(\frac{10+\sqrt{100-x^{2}}}{10}\right)-\sqrt{100-x^{2}} $$ over the interval \((0,10] .\) This graph is called a tractrix or pursuit curve. Use your school's library, the Internet, or some other reference source to find information about a tractrix. Explain how such a curve can arise in a real-life setting.
