91Ó°ÊÓ

In Problems \(7-12\), rewrite the given logarithmic expression as an equivalent exponential expression. $$ \log _{b} u=v $$

Sketch the graph of the given function \(f\). Find the \(y\) -intercept and the horizontal asymptote of the graph. State whether the function is increasing or decreasing. $$ f(x)=-1+e^{x-3} $$

In Problems \(7-12\), rewrite the given logarithmic expression as an equivalent exponential expression. $$ \log _{b} b^{2}=2 $$

Solve the given exponential equation. $$ 7^{-x}=9 $$

Sketch the graph of the given function \(f\). Find the \(y\) -intercept and the horizontal asymptote of the graph. State whether the function is increasing or decreasing. $$ f(x)=-3-e^{x+5} $$

In Problems \(13-18\), find the exact value of the given logarithm. $$ \log _{10}(0.0000001) $$

Find an exponential function \(f(x)\) \(=b^{x}\) such that the graph of \(f\) passes through the given point. $$ (3,216) $$

Iodine-131, used in nuclear medicine procedures, is radioactive and has a half-life of 8 days. Find the decay constant \(k\) for iodine-131. If the amount remaining of an initial sample after \(t\) days is given by the exponential model \(A(t)=A_{0} e^{k t},\) how long will it take for \(95 \%\) of the sample to decay?

Solve the given exponential equation. $$ 3^{2(x-1)}=7^{2} $$

Find an exponential function \(f(x)\) \(=b^{x}\) such that the graph of \(f\) passes through the given point. $$ (-1,5) $$