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Section 3.1 Exponential Functions

Example 3.2.

Bill’s parents started a college fund for Bill. When he was born they put $1,000 in a jar for Bill. Each year they add 50% of the jar’s current value into the jar. Write a formula for the amount of money, A(t), in the jar when Bill is \(t\) years old.
Solution.
After 1 year, Bill’s parents will add 50% of the original $1,000 into the jar. Thus, we have
\begin{align*} A(1)\amp =1500\\ \amp=1000+500\\ \amp =1000(1)+1000(.5)\\ \amp=\alert{1000(1.5)^1}. \end{align*}
After 2 years, Bill’s parents add 50% of the $1500 in the jar. We have
\begin{align*} A(2) \amp =\alert{1000(1.5)^1}+\alert{1000(1.5)^1}(.5)\\ \amp =\alert{1000(1.5)^1}(1.5)\\ \amp =1000(1.5)^2. \end{align*}
This trend will continue so that the amount of money in the jar after \(t\) years is given by the formula
\begin{equation*} A(t)=1000(1.5)^t. \end{equation*}
The above example was an example of an exponential function. We define such a function as follows:

Exponential Function.

\begin{equation*} P(t) = a(b)^t \text{, where } b \gt 0 \text{ and } b \ne 1 \text{, } a \ne 0. \end{equation*}
The constant \(a \) is the \(y\)-value of the \(y\)-intercept of the function.
Some examples of exponential functions are
\begin{equation*} f (x) = 5^x \text{, } P(t) = 250(1.7)^t \text{, and } g(n) = 2.4(0.3)^n. \end{equation*}
The constant \(a\) is the \(y\)-intercept of the graph because
\begin{equation*} f (0) = a \cdot b^0 = a \cdot 1 = a. \end{equation*}
For the examples above, we find that the \(y\)-intercepts are
\begin{align*} f(0) \amp= 5^0 = 1 \text{,}\\ P(0) \amp= 250(1.7)^0 = 250\text{, and}\\ g(0) \amp= 2.4(0.3)^0 = 2.4. \end{align*}
The positive constant \(b\) is called the base or the growth factor of the exponential function. We can also define the growth rate of the function to be the quantity \(r=b-1\text{,}\) which is usually expressed as a percentage value. Note that \(r\) can be positive or negative, and represents the percent change in the value of \(f(x)\) between any two \(x\)-values spaced one unit apart.
  • We do not allow \(b\) to be negative, because if \(b \lt 0\text{,}\) then \(b^x\) is not a real number for some values of \(x\text{.}\) For example, if \(b = -4\) and \(f (x) = (-4)^x\text{,}\) then \(f (1/2) = (-4)^{1/2}\) is a complex number.
  • Note that if \(b = 1\) is the growth factor, then \(1^x = 1\) for all values of \(x\text{;}\) hence the function \(f (x) = 1^x\) is actually the constant function \(f (x) = 1\text{.}\) Thus, constant functions are exponential functions!
In general, exponential functions have the following properties.

Properties of Exponential Functions, \(f(x) = a(b)^x\text{,}\) \(a \gt 0\).

  1. Domain: all real numbers.
  2. Range: all positive numbers.
  3. If \(b \gt 1\) (equivalently \(r\gt 0\)), the function is increasing;
    if \(0 \lt b \lt 1\) (equivalently \(-1\lt r\lt 0\)), the function is decreasing.
  4. The \(y\)-intercept is \((0, a)\text{.}\) There is no \(x\)-intercept.
If \(b \gt 1\) (equivalently \(r\gt 0\)), the function is said to have exponential growth . If \(0 \lt b \lt 1\) (equivalently \(-1\lt r\lt 0\)), the function is said to have exponential decay .

Subsection 3.1.1 Exponential Equations

An exponential equation is one in which the variable is part of an exponent. For example, the equation
\begin{equation*} 3^x = 81 \end{equation*}
is exponential. Many exponential equations can be solved by writing both sides of the equation as powers with the same base. To solve the equation above, we write
\begin{equation*} 3^x = 3^4 \end{equation*}
which is true if and only if \(x = 4\text{.}\) In general, if two equivalent powers have the same base, then their exponents must be equal also, as long as the base is not \(0\) or \(\pm 1\text{.}\)
Sometimes the laws of exponents can be used to express both sides of an equation as single powers of a common base.

Example 3.3.

Solve the following equations.
  1. \(\displaystyle 3^{x-2} = 9^3\)
  2. \(\displaystyle 27 \cdot 3^{-2x} = 9^{x+1}\)
Solution.
  1. Using the fact that \(9 = 3^2\text{,}\) we write each side of the equation as a power of \(3\text{:}\)
    \begin{align*} 3^{x-2} \amp = 9^3\\ 3^{x-2} \amp = \left(3^2\right)^3\\ 3^{x-2} \amp = 3^6. \end{align*}
    Now we equate the exponents to obtain
    \begin{align*} x - 2 \amp = 6\\ x \amp = 8. \end{align*}
  2. We write each factor as a power of \(3\text{.}\)
    \begin{equation*} \begin{aligned} 27 \cdot 3^{-2x} \amp= 9^{x+1} \\ 3^3 \cdot 3^{-2x} \amp= \left(3^2\right)^{x+1} \end{aligned} \end{equation*}
    We use the laws of exponents to simplify each side:
    \begin{equation*} 3^{3-2x} = 3^{2x+2}. \end{equation*}
    Now we equate the exponents to obtain
    \begin{align*} 3 - 2x \amp = 2x + 2\\ -4x =\amp -1. \end{align*}
    The solution is \(x = \dfrac{1}{4}\text{.}\)

Exercises 3.1.2 Exercises

1. Comparing Exponential Growth.

Warm-Up: Suppose you have 120 dollars to deposit into a savings account. If you put your money into Bank A, they will deposit an additional 12 dollars per year into your account. If you put your money into Bank B, they will increase your balance by 10 percent per year.
How much money would you have after one year if you put your money into Bank A? How about Bank B?
Bank A: dollars
Bank B: dollars
After two years?
Bank A: dollars
Bank B: dollars
After three years?
Bank A: dollars
Bank B: dollars
Which bank should you use?
Hint.
For Bank A, you should add 12 dollars each year.
For Bank B, you should add 10% of the value in the account from the previous year. For example, if you deposit 100 dollars in the bank, after one year you would add \(100\cdot 0.10=10\) dollars to the account. Don’t forget to move the decimal place on the percentage before multiplying!
You’ll want to choose the bank which gives you more money in the long run!

2. Writing Exponential Functions.

Mildred collects hats. In 2014, she began with 20 hats, and she plans to increase her collection by 32 percent per year. Write a formula for \(h(t)\text{,}\) the number of hats Mildred will have \(t\) years after 2014.
\(h(t) =\)
How large will her collection be in 2020? Be sure to consider what numerical values make sense in the context of the question.
Hint.
Recall the formula for an exponential function:
\begin{equation*} y=a(b)^t, \end{equation*}
where \(a\) is the initial value and \(b\) is the growth factor. We know that \(b=1+r\text{,}\) where \(r\) is the growth rate.
To find how large her collection will be in 2020, we need to evaluate \(h(6)\text{,}\) since 2020 is 6 years after 2014.

3. Properties of Exponential Functions.

The grades of six students are given by the equations below, where time is measured in weeks since the first midterm exam.
(1) \(P=85(0.92)^t\)
(2) \(P=68(.95)^t\)
(3) \(P=85(1.001)^t\)
(4) \(P=75(1.005)^t\)
(5) \(P=93(1.03)^t\)
(6) \(P=72(0.85)^t\)
a) Alex’s grade has been decreasing since the first exam. Which formula(s) could represent her grade?
b) Which student’s grade is falling the fastest?
c) Umar’s grade has been rising since the first exam; Which formula(s) could represent his grade?
d) Which student’s grade is rising the fastest?
e) Karan came out of the first exam with a 85% in the class; which formula(s) could represent his grade?
Hint.
a) Formulas that have a decreasing value would be formulas with a growth factor less than 1.
b) The fastest falling grade has the smallest growth factor.
c) Formulas that have an increasing rate would be formulas with a growth factor greater than 1.
d) The fastest rising grade has the largest growth factor.
e) The y-intercept of the formulas represents the grades earned on the first midterm. Check the initial value of each formula.

4. Writing Exponential Functions.

Suppose an investment of 600 dollars earns 10 percent interest per year. Come up with a formula for how much money you will have in \(t\) years.
\(A(t)\) =
Hint.
Recall the formula for an exponential function
\begin{equation*} y=a(b)^t, \end{equation*}
where \(a\) is the initial value and \(b\) is the growth factor. We know that \(b=1+r\text{,}\) where \(r\) is the growth rate.

5. Writing Exponential Functions.

In 2014, the worldwide giant panda population was believed to be 1835, indicating a size increase of approximately 1.9 percent per year since the last survey, which was conducted in 2003. Supposing this growth rate remains constant, write an equation relating the population, \(P\text{,}\) of giant pandas to \(t\text{,}\) the number of years since 2014.
\(P(t) =\)
Hint.
Recall the exponential formula
\begin{equation*} y=a(b)^t, \end{equation*}
where \(a\) is the initial value and \(b\) is the growth factor. We know that \(b=r+1\text{,}\) where \(r\) is the growth rate.

6. Exponential Growth.

In 2000, the population of Gotham City was 2.467 million. After 10 years, the population had increased by 16.5 percent.
a) What was the population in 2010?
million
b) Assuming a constant growth factor, by what percent did the population of Gotham City grow each year?
percent
Hint.
a) We want to take 16.5 percent of 2.467 million, since the population increased by 16.5 percent after 10 years.
b) The percent grown per year can be found by solving \(P(10) = 2.467 *(1+r)^{10}\) for \(r\text{,}\) after plugging in your answer to part a). Recall that to solve something like \(8=x^3\text{,}\) we would need to take the third root (or raise both sides to the \(\dfrac{1}{3}\) power).
We want your answer to be a percentage, so make sure you convert your final answer.

7. Properties of Exponential Growth.

Sally makes a deal with her mother. If Sally does all her chores today, she will get half a cookie. If she does her chores on the second day as well, she will get one cookie. If she does her chores on the third day, she will get two cookies. Each day in a row that she does her chores, Sally will get double the amount of cookies as the previous day. Let \(C(n)\) give the number of cookies Sally gets on day \(n\text{.}\)
a) What is the growth factor?
b) What is the initial value?
c) Write a formula for \(C(n)\text{.}\)
d) How many cookies does Sally get on day 30?
Hint.
a) Recall the growth factor is given by the constant we multiply for each increase in input. Try to make a table of values to figure out the growth factor.
b) Use the information that on Day 1, Sally gets half a cookie. That means \(C(1)=\dfrac{1}{2}\text{.}\) How would you figure out how much cookie she would earn the day before?
c) Recall the exponential formula
\begin{equation*} y=a(b)^n, \end{equation*}
where \(a\) is the initial value and \(b\) is the growth factor. Use the information from parts a) and b) to write your formula.
d) Substitute 30 in for \(n\) in the formula obtained in part c).
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