An environmental scientist is studying the rate of population growth of a species of tree in a geographic area by making yearly observations of new seedlings and newly fallen trees. She observes that the rate of growth is increasingly dramatic, approximately doubling every year.
as the height of each data point is double the height of the preceding point, in line with the “doubling every year” observation of the researcher, and we can model these data points with the formula
As the constant \(c\) merely represents a vertical scaling and/or reflection (if \(c\) is negative), we will focus our further investigation of exponential functions on the case \(c = 1\text{.}\)
Pattern9.2.3.Horizontal transformations of exponential graphs.
Given two positive bases \(a\) and \(b\text{,}\) the exponential function \(b^t\) is a horizontal transformation of the exponential function \(a^t\text{:}\)
The transformation in Pattern 9.2.3 will involve a horizontal reflection precisely when one of the bases is greater than one and the other is less than one. That is, when one exponential function is a growth function and the other is a decay function.
Just like we use \(\pi\) to represent the special number that corresponds to the area of a unit circle, the special number so that \(\ln (a) = 1\) is given its own symbol.
As \(a = e\) is the special value that makes (✶) work, it becomes “natural” to compare every exponential function to the exponential function with base \(e\text{.}\)
Subsection9.3.2The natural exponential function as a reflection of the natural logarithm
It is possible to define the natural exponential function without using exponential notation — this will help us to make sense of how to compute an exponential function’s output for irrational inputs. Consider what happens when we reflect the graph of the natural logarithm in the line \(y = t\text{.}\)
This reflected curve passes the Vertical Line Test, and so it is the graph of some function. Indeed, it looks like the graph of an exponential function, but which one? Let’s explore by plotting some points. A graph \(y = f(t)\) is a collection of input-output pairs \((t,y)\text{.}\) Reflecting in the line \(y = t\) interchanges the coordinates in a point, so that if \((t,y)\) is on the graph \(y = \ln (t)\text{,}\) then \((y,t)\) is on the reflected curve. For example, from Pattern 8.2.1, we have
\begin{gather*}
\ln (1) = 0 \\
\implies (1,0) \text{ on graph } y = \ln (t) \\
\implies (0,1) \text{ on reflected curve } \text{.}
\end{gather*}
Now, the point \((0,1)\) is on the graph of every exponential function, as \(a^0 = 1\) for each positive base \(a\text{.}\) So let’s consider some other points. Using Definition 9.3.1 along with Pattern 8.2.10, we have
\begin{equation*}
\ln (e^r) = r \ln (e) = r \cdot 1 \text{,}
\end{equation*}
so
\begin{gather*}
\ln (e^r) = r \\
\implies (e^r,r) \text{ on graph } y = \ln (t) \\
\implies (r,e^r) \text{ on reflected curve } \text{.}
\end{gather*}
Since \((r,e^r)\) is on the reflected curve for all values of \(r\text{,}\) we can say that this is indeed the graph of the natural exponential function.
Pattern9.3.7.The natural exponential function as a reflection of the natural logarithm function.
The natural exponential function \(\exp(t) = e^t\) is the unique function whose graph is the reflection in the line \(y = t\) of the graph of the natural logarithm function \(\ln (t)\text{.}\)
Subsection9.3.3Computing values of exponential functions
Realizing the natural exponential function as a reflection of the natural logarithm function, as in Pattern 9.3.7, tells us (in principle) how to compute expressions like \(e^\pi\text{.}\)
Pattern 9.3.9 says that computing \(e^\pi\) is equivalent to solving \(\ln (z) = \pi\) for \(z\text{.}\) In the Sage cell below, change the value of the z variable (which is used as the upper bound of the integration command) and re-evaluate until you get reasonably close to the value of \(\pi\text{.}\) (Recall that \(\ln (t)\) is defined as an integral of rate function \(r(t) = 1/t\text{.}\)) The first command in the Sage cell below prints out a numerical approximation of \(\pi\text{.}\)
Use the Sage cell below to do just that. As before, you are attempting to refine the value of the z variable until the integration result is very close to \(\pi \ln (2)\text{,}\) which the first command computes for you.
Subsection9.3.4Undoing the natural exponential and logarithm functions
The natural exponential and logarithm functions are not only graphical reflections of each other, they “reverse” each other algebraically, just as cube and cube root reverse each other.
We can use the patterns above to solve equations involving \(\exp\) and \(\ln\text{.}\) Before we do examples, we continue the analogy with solving equations involving cube and cube root. Suppose you wanted to solve the equation
The first step is to “undo” the cube root operation by cubing, to free the variable \(t\) from being trapped inside the cube root operation. But to maintain the equality, we must cube both sides:
\begin{gather*}
{(\sqrt[3]{t + 1})}^3 = 2^3 \\
t + 1 = 8 \\
t = 7 \text{.}
\end{gather*}
Since every exponential is really just a horizontal transformation of the natural one, we can use the same technique to solve an equation involving any exponential.
Every exponential function is a horizontal scaling (possibly including a reflection) of the natural exponential function. (See Pattern 9.3.4.) In the reflection in the line \(y = t\) between the graphs of \(\exp(t)\) and \(\ln(t)\text{,}\) a horizontal scaling of \(\exp(t)\) corresponds to a vertical scaling of \(\ln(t)\text{.}\)
a horizontal scaling by scale factor \(\ln(a)\text{.}\) In the case that \(a \gt e\) (so that \(\ln(a) \gt 1\)), the scaling will be a compression, as pictured in Figure 9.4.1. This corresponds to a vertical compression of the graph of \(\ln(t)\text{,}\) by the same scale factor. But to achieve a vertical compression, we must divide instead of multiply. Therefore, the scaled version of the natural logarithm pictured in Figure 9.4.1 is
Every logarithm function enjoys the same properties as the natural logarithm described in Section 8.2. And the relationship between logarithm and exponential of the same base also carries over.
Section9.5More properties of exponential functions
Now that we have established the “reflection” relationship between the natural exponential and logarithm functions, we can “reflect” the properties from Section 8.2 to obtain further properties of exponential functions.
From Pattern 9.3.9, \(z = e^{(u + v) \bbrac{\ln (a)}} \) should be the unique value so that \(\ln (z) = (u + v) \bbrac{\ln (a)}\text{.}\) Let’s check whether applying the natural logarithm to \(a^u a^v\) evaluates to the same result. Using the properties of the natural logarithm, we have
The “reflection” here is the interchanging of the words sum and product. Try to identify the “reflection” in each of the next two properties, compared to the corresponding property in Section 8.2. Also try to provide a justification for each of these two properties in the same manner as the justification for Pattern 9.3.4.
Pattern 9.2.2 could also be considered a “reflection” of the corresponding logarithmic pattern Pattern 8.2.3. When we reflect in the line \(y = t\text{,}\) we reflect the horizontal axis onto the vertical axis, and vice versa. So inputs become outputs and outputs become inputs. If we view the pattern
Suppose we have an exponential rate function, \(r(t) = e^t\text{.}\) This means that the rate of variation of some quantity is experiencing exponential growth, similar to the rate doubles every year pattern in Example 9.1.1. Let’s see what kind of accumulation values we get for
So it appears that an exponential rate of variation leads to exponential accumulation. The extra “minus one” is essentially a correction for the fact that an exponential function has initial value \(e^0 = 1\text{,}\) whereas accumulation should always be zero if no time has elapsed.
