Chapter 4

Solve this equation with three decimal place accuracy: $$ \log _{5}(x+5)-\log _{2} x=2 \log _{10}\left(x^{2}+2 x\right) $$

Use a graphing utility to draw the graphs of \(y=\ln \left(1+x^{2}\right)\) and \(y=\frac{1}{x}\) on the same axes. Do these graphs intersect?

Make a table for the quantities \((\sqrt{n})^{\sqrt{n+1}}\) and \((\sqrt{n+1})^{\sqrt{n}}\), with \(n=8,9,12,20,25,31,37\), \(38,43,50,100\), and 1,000 . Which of the two quantities seems to be larger? Do you think this inequality holds for all \(n \geq 8\) ?