Determine whether the following statements are true and give an explanation or counterexample.


$2=10^{{\log }_{10}2}$

Determine whether the following statements are true and give an explanation or counterexample.

$2=\ln 2^e$

Determine whether the following statements are true and give an explanation or counterexample.

${\log }_5{4}^6=4{\log }_{5}6$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f\left(x\right)=\frac{1}{x}$ , then $f^{-1}\left(x\right)=\frac{1}{x}$

Determine whether the following statements are true and give an explanation or counterexample.

$\frac{{\log }_{b}x}{{\log }_{b}y}={\log }_{b}x-{\log }_{b}y$

In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with points $P$ and $Q$ on the graph of the function.

a. Sketch a graph of the function and the secant line through $P$ and $Q$.

b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer.

After $t$ seconds, an object dropped from rest falls a distance $d=16t^2$, where $d$ is measured in feet and $2\(\leq{t}\]\leq{5}\)$.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f\left(x\right)=x^2+1$ , then $f^{-1}\left(x\right)=\frac{1}{x^2+1}$.