In Section 7.1, we studied functions that describe exponential growth or decay. A population of rabbits grew according to the formula \(~P(t) = 20(1.26)^t~\text{,}\) and a butterfly count decreased by \(~ B(t) = 3600 (0.88)^t~\text{.}\) These functions are examples of the family of exponential functions. Here are some more examples.
The positive constant \(b\) is called the base of the exponential function. The constant \(a\) gives the \(y\)-intercept, \((0,a)\text{,}\) 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
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 an imaginary number.
We also exclude \(b = 1\) as a base because \(1^x = 1\) for all values of \(x\text{;}\) hence the function \(f (x) = 1^x\) is actually the constant function \(f (x) = 1\text{.}\)
We let \(P\) represent the number of fleas present after \(t\) days. The original population of 10 fleas doubles not every day, but every 5 days. A table of values for \(P\) would look like this.
Notice that the original population is multiplied by another factor of 2 every 5 days. We must divide \(t\) by 5 to see how many times the population doubles. The formula for \(P\) is thus
We evaluate the function for \(t=\alert{7}\text{.}\) To follow the order of operations, we simplify the power before multiplying by 10. \(~\alert{\text{[TK]}}\)
(See Note 7.1.5 for another example of this calculation.) From this form see that the growth factor for this function is \(2^{1/5}\text{,}\) or about 1.149. The flea population grows at about 14.9% per day.
During an advertising campaign in a large city, the makers of Chip-O’s corn chips estimate that the number of people who have heard of Chip-O’s increases by a factor of 8 every 4 days.
If 100 people are given trial bags of Chip-O’s to start the campaign, write a function, \(N(t)\text{,}\) for the number of people who have heard of Chip-O’s after \(t\) days of advertising.
The graphs of exponential functions have two characteristic shapes, depending on whether the base, \(b\text{,}\) is greater than \(1\) or less than \(1\text{.}\) As typical examples, consider the graphs of \(f (x) = 2^x\) and \(g(x) = \left(\dfrac{1}{2}\right)^x\) shown below. Some values for \(f\) and \(g\) are recorded in the tables.
Notice that \(f(x) = 2^x\) is an increasing function and \(g(x) = \left(\dfrac{1}{2}\right)^x\) is a decreasing function. Both are concave up. In general, exponential functions have the following properties.
In the table for \(f(x)\text{,}\) you can see that as the \(x\)-values decrease toward negative infinity, the corresponding \(y\)-values decrease toward zero. As a result, the graph of \(f\) decreases toward the \(x\)-axis, but never touches it, as we move to the left.
How does the value of \(b\) affect the graph? For increasing functions, the larger the value of the base, \(b\text{,}\) the faster the function grows. In the Example below, we compare two exponential functions with different bases.
Then we plot the points for each function and connect them with smooth curves. Note that \(a=1\) for both functions, so their graphs have the same \(y\)-intercept, \((0,1)\text{.}\)
For positive \(x\)-values, \(g(x)\) is always larger than \(f(x)\text{,}\) and is increasing more rapidly. In the figure, we can see that \(g(x) = 4^x\) climbs more rapidly than \(f(x) = 3^x\text{.}\) However, for negative \(x\)-values, \(g(x)\) is smaller than \(f(x)\text{.}\)
Perhaps you see a similarity in the formula for exponential functions, \(y = ab^x\text{,}\) and the formula for linear functions, \(y = mx + b\text{.}\) In an exponential function the base \(b\) tells us how fast the function increases or decreases, much as the slope \(m\) does for linear functions, and the constant \(a\) tells us the intial value when \(x=0\text{,}\) as does the constant \(b\) in the linear formula \(y = mx + b\text{.}\)
Subsection7.2.3Comparing Exponential and Power Functions
We have studied several families of functions, including linear, quadratic, and power functions. Exponential functions are useful because they model growth or decline by a constant factor.
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.
We can tell that \(g\) is the exponential function because its values increase by a factor of 3 for each unit increase in \(x\text{.}\) The base of the function is \(b=3\text{.}\) We also see from the table that \(a=g(0)=2\text{,}\) so \(g(x)=2(3^x)\text{.}\) Thus, \(f\) must be the power function, \(f(x)=kx^3\text{.}\) To find \(k\text{,}\) we notice that \(f(1)=k\text{,}\) so \(k=2\text{,}\) and \(f(x)=2x^3\text{.}\)
As you can see from the figure, the graphs of the two functions in the previous Example are also quite different. In particular, note that the power function passes through the origin, while the exponential function approaches the negative \(x\)-axis as a horizontal asymptote.
From the table, we see that \(f(3) = g(3) = 54\text{,}\) so the two graphs intersect at \(x = 3\text{.}\) (They also intersect at approximately \(x = 2.48\text{.}\)) However, if you compare the values of \(f(x) = 2x^3\) and \(g(x) = 2(3^x)\) for larger values of \(x\text{,}\) you will see that eventually the exponential function overtakes the power function.
The relationship in Example 7.2.13 holds true for all increasing power and exponential functions: For large enough values of \(x\text{,}\) the exponential function will always be greater than the power function, regardless of the parameters in the functions. The figure at left shows the graphs of \(f(x) = x^6\) and \(g(x) = 1.8^x\text{.}\) At first, \(f(x) \gt g(x)\text{,}\) but at around \(x = 37\text{,}\)\(g(x)\) overtakes \(f(x)\text{,}\) and \(g(x) \gt f(x)\) for all \(x \gt 37\text{.}\)
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 \(\pm 1\text{.}\)
In Example 1 we wrote a formula for a population of fleas that started with 10 fleas and doubles in number every 5 days. How long will it be before there are 10,240 fleas?
We set \(P = \alert{10,240}\) and solve for \(t\text{:}\)
\begin{align*}
\alert{10,240} \amp = 10\cdot 2^{t/5}\amp\amp \blert{\text{Divide both sides by 10.}}\\
1024 \amp = 2^{t/5} \amp\amp \blert{\text {Write 1024 as a power of 2.}}\\
2^{10} \amp = 2^{t/5}
\end{align*}
We equate the exponents to get \(10 = \dfrac{t}{5}\text{,}\) or \(t = 50\text{.}\) The population will grow to 10,240 fleas in 50 days.
Subsection7.2.5Graphical Solution of Exponential Equations
It is not always so easy to express both sides of the equation as powers of the same base. In the following sections, we will develop more general methods for finding exact solutions to exponential equations. But we can use a graphing utility to obtain approximate solutions.
Using the TRACE feature, we see that the \(y\)-coordinates are too small when \(x \lt 2.1\) and too large when \(x \gt 2.4\text{.}\) The solution we want lies somewhere between \(x = 2.1\) and \(x = 2.4\text{,}\) but this approximation is not accurate enough.
To improve our approximation, we can use the intersect feature because the \(x\)-coordinate of the intersection point of the two graphs is the solution of the equation \(2^x = 5\text{.}\) Activating the intersect command results in figure (b), and we see that, to the nearest hundredth, the solution is \(2.32\text{.}\)
Evaluating \(2^{2.32}\) yields \(4.993322196\text{.}\) This number is not equal to \(5\text{,}\) but it is close, so we believe that \(x = 2.32\) is a reasonable approximation to the solution of the equation \(2^x = 5\text{.}\)
In Practice 1 we wrote a function, \(N(t) = 100 \cdot 8^{t/4}\text{,}\) for the number of people who have heard of Chip-O’s after \(t\) days of advertising.
Use your grapher to graph the function \(N(t)\) for \(0 \le t \le 15\text{.}\)
For Problems 19 and 20, graph each pair of functions on the same axes by making a table of values and plotting points by hand. Choose appropriate scales for the axes.
For Problems 26 and 27, which tables could describe exponential functions? Explain why or why not. If the function is exponential, find its growth or decay factor.
The frequency of a musical note depends on its pitch. The graph shows that the frequency increases exponentially. The function \(F(p) = F_0b^p\) gives the frequency as a function of the number of half-tones, \(p\text{,}\) above the starting point on the scale
Find an approximation for the growth factor, \(b\text{,}\) by comparing two points on the graph. (Some of the points on the graph of \(F(p)\) are approximately \((1, 466)\text{,}\)\((2, 494)\text{,}\)\((3, 523)\text{,}\) and \((4, 554)\text{.}\))
The frequency doubles when you raise a note by one octave, which is equivalent to 12 half-tones. Use this information to find an exact value for \(b\text{.}\)
For several days after the Northridge earthquake on January 17, 1994, the area received a number of significant aftershocks. The red graph shows that the number of aftershocks decreased exponentially over time. The graph of the function \(S(d) = S_0b^d\text{,}\) shown in black, approximates the data. (Source: Los Angeles Times, June 27, 1995)
Find an approximation for the decay factor, \(b\text{,}\) by comparing two points on the graph. (Some of the points on the graph of \(S(d)\) are approximately \((1, 82)\text{,}\)\((2, 45)\text{,}\)\((3, 25)\text{,}\) and \((4, 14)\text{.}\))
Related species living in the same area often evolve in different sizes to minimize competition for food and habitat. Here are the masses of eight species of fruit pigeon found in New Guinea, ranked from smallest to largest. (Source: Burton, 1998)
Compute the ratios of the masses of successive sizes of fruit pigeons. Are the ratios approximately constant? What does this information tell you about your answer to part (a)?