Exponential growth is modeled by increasing functions of the form \(P(t)=P_0b^t\text{,}\) where the growth factor, \(b\text{,}\) is a number greater than 1.
Exponential Growth.
The function
\begin{equation*}
P(t)=P_0 b^t
\end{equation*}
describes exponential growth, where \(P_0=P(0)\) is the initial value of the function and \(b\) is the growth factor.
describes exponential growth at a constant percent rate of growth, \(r\text{.}\)
The initial value of the function is \(P_0 = P(0)\text{,}\) and \(b=1+r\) is the growth factor.
Compound Interest.
If a principal of \(P\) dollars is invested in an account that pays an interest rate \(r\) compounded annually, the balance \(B\) after \(t\) years is given by
If \(0 \lt b \lt 1\text{,}\) then \(P(t)=P_0b^t\) is a decreasing function. In this case \(P(t)\) is said to describe exponential decay.
A percent increase of \(r\) (in decimal form) corresponds to a growth factor of \(b=1+r\text{.}\) A percent decrease of \(r\) corresponds to a decay factor of \(b=1-r\text{.}\)
Exponential Growth and Decay.
The function
\begin{equation*}
P(t) = P_0 b^t
\end{equation*}
models exponential growth and decay.
\(P_0 =P(0)\) is the initial value of \(P\text{;}\)
\(b\) is the growth or decay factor.
If \(b \gt 1\text{,}\) then \(P(t)\) is increasing, and \(b = 1 + r\text{,}\) where \(r\) represents percent increase.
If \(0 \lt b \lt 1\text{,}\) then \(P(t)\) is decreasing, and \(b = 1 - r\text{,}\) where \(r\) represents percent decrease.
The growth factor of an exponential function is analogous to the slope of a linear function: Each measures how quickly the function is increasing (or decreasing).
Suppose \(L(t)=a+mt\) is a linear function and \(E(t)=ab^t\) is an exponential function. For each unit that \(t\) increases, \(m\) units are added to the value of \(L(t)\text{,}\) whereas the value of \(E(t)\) is multiplied by \(b\text{.}\)
We do not allow the base of an exponential function to be negative, because if \(b \lt 0\text{,}\) then \(b^x\) is not a real number for some values of \(x\text{.}\)
Properties of Exponential Functions, \(f(x) = ab^x\text{,}\)\(a \gt 0\).
If \(b \gt 1\text{,}\) the function is increasing and concave up;
if \(0 \lt b \lt 1\text{,}\) the function is decreasing and concave up.
The \(y\)-intercept is \((0,a)\text{.}\) There is no \(x\)-intercept.
The negative \(x\)-axis is a horizontal asymptote for exponential functions with \(b \gt 1\text{.}\) For exponential functions with \(0 \lt b \lt 1\text{,}\) the positive \(x\)-axis is an asymptote.
Exponential functions are not the same as the power functions we studied earlier. Although both involve expressions with exponents, it is the location of the variable that makes the difference.
Power Functions vs Exponential Functions.
\(\hphantom{General formula and m}\)
Power Functions
Exponential Functions
General formula
\(h(x)=kx^p\)
\(f(x)=ab^x\)
Description
variable base and constant exponent
constant base and variable exponent
Example
\(h(x)=2x^3\)
\(f(x)=2(3^x)\)
Many exponential equations can be solved by writing both sides of the equation as powers with the same base. If two equivalent powers have the same base, then their exponents must be equal also (as long as the base is not 0 or \(\pm 1\)).
Definition of Logarithm.
For \(b\gt 0, b\ne 1\text{,}\) the base \(b\) logarithm of \(x\), written \(\log_{b} x\text{,}\) is the exponent to which \(b\) must be raised in order to yield \(x\text{.}\)
Logarithms and Exponents: Conversion Equations.
If \(b \gt 0\text{,}\)\(b\ne 1\text{,}\) and \(x \gt 0\text{,}\)
\begin{equation*}
\blert{y = \log_b x}~~~ \text{ if and only if }~~~ \blert{ x = b^y}
\end{equation*}
Some Useful Logarithms.
For any base \(b \gt 0, b\ne 1\text{,}\)
\begin{align*}
\log_b b \amp = 1~~~ \text{ because } ~~~b^1 = b\\
\log_b 1 \amp = 0 ~~~ \text{ because } ~~~b^0 = 1\\
\log_b{b^x} \amp = x~~~ \text{ because } ~~~b^x = b^x
\end{align*}
We use logarithms to solve exponential equations, just as we use square roots to solve quadratic equations. The operation of taking a base \(b\) logarithm is the inverse of raising the base \(b\) to a power, just as extracting square roots is the inverse of squaring a number.
Steps for Solving Base 10 Exponential Equations.
Isolate the power on one side of the equation.
Rewrite the equation in logarithmic form.
Use a calculator, if necessary, to evaluate the logarithm.
Solve for the variable.
Properties of Logarithms.
If \(x,~y,\) and \(b\gt 0,\) and \(b\ne 1\text{,}\) then
Take the base 10 logarithm of both sides of the equation.
Apply the third property of logarithms to simplify.
Use a calculator, if necessary, to evaluate the logarithm.
Solve for the variable.
Exercises7.6.3Chapter 7 Review Problems
Exercise Group.
For Problems 1–4,
Write a function that describes exponential growth or decay.
Evaluate the function at the given values.
1.
The number of computer science degrees awarded by Monroe College has increased by a factor of 1.5 every 5 years since 1984. If the college granted 8 degrees in 1984, how many did it award in 1994? In 2005?
Answer.
\(\displaystyle D = 8(1.5)^{t/5}\)
\(18\text{;}\)\(44\)
2.
The price of public transportation has been rising by 10% per year since 1975. If it cost $0.25 to ride the bus in 1975, how much did it cost in 1985? How much will it cost in the year 2030 if the current trend continues?
3.
A certain medication is eliminated from the body at a rate of 15% per hour. If an initial dose of 100 milligrams is taken at 8 a.m., how much is left at 12 noon? At 6 p.m.?
Answer.
\(\displaystyle M = 100(0.85)^t\)
\(52.2\) mg; \(19.7\) mg
4.
After the World Series, sales of T-shirts and other baseball memorabilia decline 30% per week. If $200,000 worth of souvenirs were sold during the Series, how much will be sold 4 weeks later? After 6 weeks?
Exercise Group.
For Problems 5–8, graph the function.
5.
\(f (t) = 6(1.2)^t\)
Answer.
6.
\(g(t) = 35(0.6)^{-t}\)
7.
\(P(x) = 2^x - 3\)
Answer.
8.
\(R(x) = 2^{x+3}\)
Exercise Group.
For Problems 9-12, solve the equation.
9.
\(3^{x+2} = 9^{1/3}\)
Answer.
\(\dfrac{-4}{3}\)
10.
\(2^{x-1} = 8^{-2x}\)
11.
\(4^{2x+1} = 8^{x-3}\)
Answer.
\(-11\)
12.
\(3^{x^2-4} = 27\)
Exercise Group.
For Problems 13-18, find the logarithm.
13.
\(\log_2 16 \)
Answer.
\(4\)
14.
\(\log_4 2 \)
15.
\(\log_3 \dfrac{1}{3} \)
Answer.
\(-1\)
16.
\(\log_7 7 \)
17.
\(\log_{10} 10^{-3} \)
Answer.
\(-3\)
18.
\(\log_{10} 0.0001 \)
Exercise Group.
For Problems 19 and 20, write the equation in exponential form.
19.
\(\log_{2} 3 = x-2\)
Answer.
\(2^{x-2} = 3\)
20.
\(\log_{n} q = p-1\)
Answer.
\(n^{p-1} = q\)
Exercise Group.
For Problems 21 and 22, write the equation in logarithmic form.
21.
\(0.3^{-2} = x + 1\)
Answer.
\(\log_{0.3}(x + 1) = -2\)
22.
\(4^{0.3 t} = 3N_0\)
Exercise Group.
For Problems 23-30, solve for the unknown value.
23.
\(\log_{3}\dfrac{1}{3}=y \)
Answer.
\(-1\)
24.
\(\log_{3}x=4 \)
25.
\(\log_{b} 16=2 \)
Answer.
\(4 \)
26.
\(\log_{2}(3x-1)=3 \)
27.
\(4\cdot 10^{1.3x} = 20.4\)
Answer.
\(\dfrac{\log 5.1}{1.3}\approx 0.5433 \)
28.
\(127= 2(10^{0.5x} )-17.3\)
29.
\(3 (10^{-0.7x})+6.1 = 9\)
Answer.
\(\dfrac{\log (2.9/3)}{-0.7}\approx 0.21 \)
30.
\(40(1-10^{-1.2x} )=30\)
Exercise Group.
In Problems 31-34, evaluate the expression.
31.
\(k=\dfrac{1}{t}(\log N - \log N_0);~~\) for \(t=2.3,~N=12,000,\) and \(N_0 =9000\)
Answer.
0.054
32.
\(P=\dfrac{1}{k} \sqrt{\dfrac{\log N}{t}}:~~\) for \(k=0.4,~N=48,\) and \(t=12\)
Answer.
2.959
33.
\(h = k \log \left(\dfrac{N}{N-N_0}\right)~~\) for \(k=1.2,~N=6400,\) and \(N_0 =2000\)
Answer.
0.195
34.
\(Q = \dfrac{1}{t}\left(\dfrac{\log M}{\log N}\right);~~\) for \(t=0.3,~M=180,\) and \(N=640\)
Answer.
2.823
Exercise Group.
For Problems 35-38, write the expression in terms of simpler logarithms. (Assume that all variables and variable expressions denote positive real numbers.)