Chapter 6

In Problems 1-20, solve the given exponential equation. 5^{x-2}=1

Use the laws of logarithms to show that for \(f(x)=\) \(\log _{b} x\) $$ \frac{f(x+h)-f(x)}{h}=\frac{1}{h} \log _{b}\left(1+\frac{h}{x}\right)=\frac{1}{x} \log _{b}\left(1+\frac{h}{x}\right)^{x / h}. $$

Graph the given functions on the same rectangular coordinate system. $$ y=3^{x}, y=3^{-x} $$

After 2 hours the number of bacteria in a culture is observed to have doubled. (a) Find an exponential model (1) for the number of bacteria in the culture at time \(t\). (b) Find the number of bacteria present in the culture after 5 hours. (c) Find the time that it takes the culture to grow to 20 times its initial size.

In Problems \(1-6\), rewrite the given exponential expression as an equivalent logarithmic expression. $$ 4^{-1 / 2}=\frac{1}{2} $$

Solve the given exponential equation. 3^{x}=27^{x}

Graph the given functions on the same rectangular coordinate system. $$ y=-2^{x}, y=-2^{-x} $$

In Problems \(1-6\), rewrite the given exponential expression as an equivalent logarithmic expression. $$ 9^{\circ}=1 $$

Solve the given exponential equation. $$ 10^{-2 x}=\frac{1}{10,000} $$

Graph the given functions on the same rectangular coordinate system. $$ y=\left(\frac{3}{4}\right)^{x}, y=\left(\frac{4}{3}\right)^{x} $$

A model for the population in a small community is given by \(P(t)=1500 e^{k t} .\) If the initial population increases by \(25 \%\) in 10 years, what will the population be in 20 years?

In Problems \(1-6\), rewrite the given exponential expression as an equivalent logarithmic expression. $$ 10^{4}=10,000 $$

Solve the given exponential equation. $$ 27^{x}=\frac{9^{2 x-1}}{3^{x}} $$

Graph the given functions on the same rectangular coordinate system. $$ y=-\left(\frac{1}{3}\right)^{x}, y=\left(\frac{1}{3}\right)^{-x} $$

In Problems \(1-6\), rewrite the given exponential expression as an equivalent logarithmic expression. $$ 10^{0.3010}=2 $$

Solve the given exponential equation. $$ e^{5 x-2}=30 $$

Graph the given functions on the same rectangular coordinate system. $$ y=3^{x-1}, y=3^{-x+1} $$

A model for the number of bacteria in a culture after \(t\) hours is given by \(P(t)=P_{0} e^{k t} .\) After 3 hours it is observed that 400 bacteria are present. After 10 hours 2000 bacteria are present. What was the initial number of bacteria?

In Problems \(1-6\), rewrite the given exponential expression as an equivalent logarithmic expression. $$ t^{-s}=v $$

Solve the given exponential equation. $$ \left(\frac{1}{e}\right)^{x}=e^{3} $$

Graph the given functions on the same rectangular coordinate system. $$ y=2^{x-2}, y=-2^{x+2} $$

In genetic research a small colony of drosophila (small two-winged fruit flies) is grown in a laboratory environment. After 2 days it is observed that the population of flies in the colony has increased to \(200 .\) After 5 days the colony has 400 flies. (a) Find a model \(P(t)=P_{0} e^{k t}\) for the population of the fruit-fly colony after \(t\) days. (b) What will be the population of the colony in 10 days? (c) When will the population of the colony be 5000 fruit flies?

Solve the given exponential equation. $$ 2^{x} \cdot 3^{x}=36 $$

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. $$ f(x)=-5+3^{x} $$

A student sick with a flu virus returns to an isolated college campus of 2000 students. A model for the number of students infected with the flu \(t\) days after the student's return is given by the logistic function $$ P(t)=\frac{2000}{1+1999 e^{-0.8905 t}} $$ (a) According to this model, how many students will be infected with the flu after 5 days? (b) How long will it take for one-half of the student population to become infected? (c) How many students does the model predict will become infected after a very long period of time? (d) Sketch a graph of \(P(t)\).

In Problems \(7-12\), rewrite the given logarithmic expression as an equivalent exponential expression. $$ \log _{2} 128=7 $$

Solve the given exponential equation. $$ \frac{4^{x}}{3^{x}}=\frac{9}{16} $$

Compute \(\frac{f(x+h)-f(x)}{h}\) for the given function. $$ f(x)=e^{-x+4} $$

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. $$ f(x)=2+3^{-x} $$

In 1920 , Raymond Pearl and Lowell Reed proposed a logistic model for the population of the United States based on the years \(1790,1850,\) and \(1910 .\) The logistic function they proposed was $$ P(t)=\frac{2930.3009}{0.014854+e^{-0.0313395 t}} $$ where \(P\) is measured in thousands and \(t\) represents the number of years past 1780 . (a) The model agrees quite well with the census figures between 1790 and \(1910 .\) Determine the population figures for \(1790,1850,\) and \(1910 .\) (b) What does this model predict for the population of the United States after a very long time? How does this prediction compare with the 2000 census population of 281 million?

In Problems \(7-12\), rewrite the given logarithmic expression as an equivalent exponential expression. $$ \log _{5} \frac{1}{25}=-2 $$

Solve the given exponential equation. $$ 2^{x}=8^{2 x-3} $$

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. $$ f(x)=3-\left(\frac{1}{5}\right)^{x} $$

Initially 200 milligrams of a radioactive substance was present. After 6 hours the mass had decreased by \(3 \%\). Construct an exponential model \(A(t)=A_{0} e^{k t}\) for the amount remaining of the decaying substance after \(t\) hours. Find the amount remaining after 24 hours.

In Problems \(7-12\), rewrite the given logarithmic expression as an equivalent exponential expression. $$ \log _{\sqrt{3}} 81=8 $$

Solve the given exponential equation. $$ \frac{1}{4}\left(10^{-2 x}\right)=25\left(10^{x}\right) $$

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. $$ f(x)=9-e^{x} $$

In Problems \(7-12\), rewrite the given logarithmic expression as an equivalent exponential expression. $$ \log _{16} 2=\frac{1}{4} $$

Solve the given exponential equation. $$ 5-10^{2 x}=0 $$

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. $$ f(x)=-1+e^{x-3} $$

Do this problem without using the exponential model (3). Initially there are 400 grams of a radioactive substance on hand. If the half-life of the substance is 8 hours, give an educated guess of how much remains (approximately) after 17 hours. After 23 hours. After 33 hours.

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

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

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. $$ (3,216) $$

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?

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

Solve the given exponential equation. $$ \left(\frac{1}{2}\right)^{-x+2}=8\left(2^{x-1}\right)^{3} $$