91Ó°ÊÓ

For the following exercises, evaluate the exponential functions for the indicated value of \(x .\) $$h(x)=-\frac{1}{2}\left(\frac{1}{2}\right)^{x}+6 \text { for } h(-7)$$

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(f(x)=a b^{x}+d .\) $$-50=-\left(\frac{1}{2}\right)^{-x}$$

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(f(x)=a b^{x}+d .\) $$116=\frac{1}{4}\left(\frac{1}{8}\right)^{x}$$

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(f(x)=a b^{x}+d .\) $$12=2(3)^{x}+1$$

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(f(x)=a b^{x}+d .\) $$5=3\left(\frac{1}{2}\right)^{x-1}-2$$

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(f(x)=a b^{x}+d .\) $$-30=-4(2)^{x+2}+2$$

Explore and discuss the graphs of \(F(x)=(b)^{x}\) and \(G(x)=\left(\frac{1}{b}\right)^{x}\) . Then make a conjecture about the relationship between the graphs of the functions \(b^{x}\) and \(\left(\frac{1}{b}\right)^{x}\) for any real number \(b>0\) .

Explore and discuss the graphs of \(f(x)=4^{x}, g(x)=4^{x-2},\) and \(h(x)=\left(\frac{1}{16}\right)^{x}\) . Then make a conjecture about the relationship between the graphs of the functions \(b^{x}\) and \(\left(\frac{1}{b^{n}}\right) b^{x}\) for any real number \(n\) and real number \(b>0\) .

What is a base \(b\) logarithm? Discuss the meaning by interpreting each part of the equivalent equations \(b^{y}=x\) and \(\log _{b} x=y\) for \(b>0, b \neq 1\)

How is the logarithmic function \(f(x)=\log _{b} x\) related to the exponential function \(g(x)=b^{x} ?\) What is the result of composing these two functions?