CCM Exponential growth

Chapter 9 Exponential growth

Section 9.1 Exponential functions

Example 9.1.1. A doubling rate of growth.

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.

Figure 9.1.2. Data points for rate of growth of a population. Note that axes are not the same scale.

To create a mathematical model from the data points, the researcher fits a curve to them.

Figure 9.1.3. Fitted curve for rate of growth of a population.

If we assign a scale on both axes in Figure 9.1.3 so that the grid lines fall at integer values, the the plotted points fall at

\begin{align*} r(0) \amp = 1 \amp r(1) \amp = 2 \amp r(2) \amp = 4 \amp r(3) \amp = 8 \amp r(4) \amp = 16\text{,} \end{align*}

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

\begin{equation*} r(t) = 2^t \text{.} \end{equation*}

If we plot some additional points, we find the fitted curve matches other values of this rate function.

Plotting additional rate function values against our "doubling" model curve. — Figure 9.1.4. Plotting function values \(r(-1) = 2^{-1}\) and \(r(1/2) = 2^{1/2}\) on our “doubling” model curve.

It appears that the model curve matches the values of the function \(r(t) = 2^t\) not only at whole, positive exponents but also at

negative exponents, which are used to mean reciprocal:
\begin{equation*} 2^{-a} = \frac{1}{2^a} \end{equation*}
rational exponents, which are used to combine a power with a root:
\begin{equation*} 2^{a/b} = \sqrt[b]{2^a} \text{.} \end{equation*}

We are not yet ready to attach a meaning to irrational exponents, such as

\begin{equation*} 2^{\pi} \text{,} \end{equation*}

but for the moment we will assume that our formula \(r(t) = 2^t \) matches our model curve at all values of \(t\text{.}\)

Definition 9.1.5. Exponential function with base \(a\).

For positive constant \(a\text{,}\) the exponential function with base \(a\) is the function

\begin{equation*} f(t) = c a^t \text{,} \end{equation*}

where the constant \(c\) represents the initial value \(c = f(0)\text{.}\)

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 \(c = 1\text{.}\)

Exponential functions come in three varieties:

exponential growth when \(a \gt 1 \)
constant when \(a = 1\)
exponential decay when \(0 \lt a \lt 1\text{.}\)

A typical exponential decay graph. — (a) A typical exponential growth graph.

Section 9.2 Basic properties of exponential functions

From the graphs in Figure 9.1.6 we have some obvious properties.

Pattern 9.2.1. Domain and range.

The domain of an exponential function covers the whole real number line, with no singular points, and the range consists of all positive real numbers.

Pattern 9.2.2. Long-term behaviour.

Growth.

When \(a \gt 1\text{,}\) we have

\begin{equation*} a^t \to \infty \quad \text{ as } \quad t \to \infty \end{equation*}

and

\begin{equation*} a^t \to 0 \quad \text{ as } \quad t \to -\infty \text{.} \end{equation*}
Decay.

When \(0 \lt a \lt 1\text{,}\) we have

\begin{equation*} a^t \to 0 \quad \text{ as } \quad t \to \infty \end{equation*}

and

\begin{equation*} a^t \to \infty \quad \text{ as } \quad t \to -\infty \text{.} \end{equation*}

The graphs of all exponential functions look essentially the same because they are all horizontal transformations of each other.

Pattern 9.2.3. 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 \(a^t\text{:}\)

\begin{equation*} b^t = a^{c t}\text{,} \qquad c = \frac{\ln(b)}{\ln(a)} \text{.} \end{equation*}

Note that if \(c\) is negative, this transformation involves not only a horizontal scale but also a reflection.

Justification.

Since the “new” base \(b\) is a positive number, it is in the range of the exponential function \(a^t\text{.}\)

Figure 9.2.4. New base as an output of an exponential function.

We can use Pattern 8.2.10 to determine the value of \(t\) so that \(a^t = b\text{:}\)

\begin{gather*} a^t = b \\ \ln (a^t) = \ln (b) \\ t \ln (a) = \ln (b) \\ t = \frac{\ln (b)}{\ln (a)} \text{.} \end{gather*}

Substituting \(b = a^{\bbrac{\ln (b) / \ln (a)}}\) into the exponential function formula \(b^t\text{,}\) we obtain

\begin{equation*} b^t = {(a^{\bbrac{\ln (b) / \ln (a)}})}^t = a^{t \bbrac{\ln (b) / \ln (a)}} \text{.} \end{equation*}

Figure 9.2.5. Graph of one exponential function as a horizontal scaling of another.

Remark 9.2.6.

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.

Section 9.3 The natural exponential function

Subsection 9.3.1 Euler’s number

In Pattern 9.2.3, if we choose \(a\) so that \(\ln (a) = 1\text{,}\) then the comparison between the two exponential formulas becomes much simpler:

\begin{gather} b^t = a^{t \ln (b)} \text{.}\tag{✶} \end{gather}

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.

Definition 9.3.1. Euler’s number.

The real number, denoted by the letter \(e\text{,}\) so that \(\ln (e) = 1 \text{.}\)

Euler’s number defined relative to a special rate function. — Figure 9.3.2. Euler’s number defined relative to the special rate function \(r(t) = 1/t\text{.}\)

Sage knows about Euler’s number e. We can express it as a decimal number using the n() method.

As \(a = e\) is the special value that makes (✶) work, it becomes “natural” to compare every exponential function to the exponential function with base \(e\text{.}\)

Definition 9.3.3. Natural exponential function.

The function \(\exp(t) = e^t\text{,}\) where the constant base \(e\) is Euler’s number.

Pattern 9.3.4. Realizing an exponential function as a horizontal transformation of the natural exponential function.

For every positive base \(a\text{,}\) we have

\begin{equation*} a^t = e^{t \ln (a)} \text{.} \end{equation*}

Subsection 9.3.2 The 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 \(y = t\text{.}\)

Figure 9.3.5. A reflection of the graph of the natural logarithm function.

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*}

\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*}

Figure 9.3.6. Specific points on a reflection of the graph of the natural logarithm function.

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.

Pattern 9.3.7. 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 \(\ln (t)\text{.}\)

Remark 9.3.8.

We will explore pairs of functions whose graphs are reflections of each other more in Chapter 16.

Subsection 9.3.3 Computing 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. Using the natural logarithm to compute values of the natural exponential function.

The following two equalities are equivalent to each other.

\begin{align*} e^w \amp = z \amp \ln (z) \amp = w \text{.} \end{align*}

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 \(\pi\text{.}\)

Of course, computers and calculators know how to (approximately) compute values of \(e^t\) for arbitrary values of \(t\text{.}\)

We can combine Pattern 9.3.9 with Pattern 9.3.4 and use a similar method to compute values like

\begin{equation*} 2^\pi = e^{\pi \ln (2)} \end{equation*}

by solving

\begin{equation*} \ln (z) = \pi \ln (2) \text{.} \end{equation*}

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.

When you’ve had enough of trial-and-error attempts to determine the value of \(2^\pi\) above, you can ask Sage to compute it for you below.

Subsection 9.3.4 Undoing 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.

Pattern 9.3.10. Composing natural exponential and logarithm.

Exponential inside logarithm.

For all \(a\text{,}\) we have

\begin{equation*} \ln (e^a) = a \text{.} \end{equation*}
Logarithm inside exponential.

For all positive \(a\text{,}\) we have

\begin{equation*} e^{\ln (a)} = a \text{.} \end{equation*}

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

\begin{equation*} \sqrt[3]{t + 1} = 2 \text{.} \end{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*}

Similarly, to solve the equation

\begin{equation*} {(t + 1)}^3 = 2 \text{,} \end{equation*}

the first step is to “undo” the cube operation on the left by taking the cube root of both sides:

\begin{gather*} \sqrt[3]{{(t + 1})^3} = \sqrt[3]{2} \\ t + 1 = \sqrt[3]{2} \\ t = \sqrt[3]{2} - 1 \text{.} \end{gather*}

We can perform similar algebra using \(\exp\) and \(\ln\text{.}\)

Example 9.3.11. Solving an equation involving the natural exponential function.

To solve the equation

\begin{equation*} e^{t + 1} = 2 \text{,} \end{equation*}

we first “undo” the exponential by taking the logarithm of both sides and applying Property 1 of Pattern 9.3.10:

\begin{gather*} \ln (e^{t + 1}) = \ln (2) \\ t + 1 = \ln (2) \\ t = \ln (2) - 1 \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.

Example 9.3.12. Solving an equation involving an exponential function.

To solve the equation

\begin{equation*} a^{t + 1} = 2 \text{,} \end{equation*}

we first take the logarithm of both sides and apply Pattern 8.2.10:

\begin{gather*} \ln (a^{t + 1}) = \ln (2) \\ (t + 1) \ln (a) = \ln (2) \\ t + 1 = \frac{\ln (2)}{\ln (a)} \\ t = \frac{\ln (2)}{\ln (a)} - 1 \text{.} \end{gather*}

Finally, an example with the roles of \(\exp\) and \(\ln\) reversed.

Example 9.3.13. Solving an equation involving the natural logarithm function.

To solve the equation

\begin{equation*} \ln (t + 1) = 2 \text{,} \end{equation*}

we first “undo” the logarithm by taking the natural exponential of both sides and applying Property 2 of Pattern 9.3.10:

\begin{gather*} e^{\ln (t + 1)} = e^2 \\ t + 1 = e^2 \\ t = e^2 - 1 \text{.} \end{gather*}

Section 9.4 Logarithms in other bases

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 \(\ln(t)\text{.}\)

Figure 9.4.1. A reflection of the graph of a horizontal scaling of the natural exponential function.

We know that

\begin{equation*} a^t = \exp\bbrac{t \ln(a)}\text{,} \end{equation*}

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

\begin{equation*} \ell(t) = \frac{\ln(t)}{\ln(a)} \text{.} \end{equation*}

Definition 9.4.2. Base-\(a\) logarithm.

For \(a \gt 0\text{,}\) define the base-\(a\) logarithm to be the function

\begin{equation*} \log_a(t) = \frac{\ln(t)}{\ln(a)} \text{.} \end{equation*}

Note 9.4.3.

Since \(\ln(e) = 1\text{,}\) a logarithm in base \(a = e\) is just the natural logarithm function \(\ln(t)\text{.}\)

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.

Pattern 9.4.4. Composing natural exponential and logarithm.

Assume \(a \gt 0\text{.}\)

Exponential inside logarithm.

For all \(t\text{,}\) we have

\begin{equation*} \log_a (a^t) = t \text{.} \end{equation*}
Logarithm inside exponential.

For all positive \(t\text{,}\) we have

\begin{equation*} a^{\log_a(t)} = t \text{.} \end{equation*}

In particular, note that \(\log_a (a) = 1 \text{.}\)

Section 9.5 More 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.

Pattern 9.5.1. Exponential of a sum.

For positive base \(a\text{,}\) we have

\begin{equation*} a^{u + v} = a^u a^v \text{.} \end{equation*}

In other words, the exponential of a sum is the product of the exponentials.

Justification.

From Pattern 9.3.4, we have

\begin{equation*} a^{u + v} = e^{(u + v) \bbrac{\ln (a)}} \text{.} \end{equation*}

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

\begin{align*} \ln (a^u a^v) \amp = \ln (a^u) + \ln (a^v) \\ \amp = u \ln (a) + v \ln (a) \\ \amp = (u + v) \ln (a) \text{,} \end{align*}

as desired.

Remark 9.5.2.

Comparing Pattern 9.5.1 with Pattern 8.2.4, the patterns are

the exponential of a sum is the product of the exponentials

versus

the logarithm of a product is the sum of the logarithms.

The “reflection” here is the interchanging of the words sum and product. See if you can see the “reflection” in each of the next two properties, compared to the corresponding property in Section 8.2. Also see if you can provide a justification for each of these two properties in the same manner as the justification for Pattern 9.3.4.

Pattern 9.5.3. Exponential of a negative.

For positive base \(a\text{,}\) we have

\begin{equation*} a^{- u} = \frac{1}{a^u} \text{.} \end{equation*}

In other words, the exponential of a negative is the reciprocal of the exponential.

Pattern 9.5.4. Exponential of a difference.

For positive base \(a\text{,}\) we have

\begin{equation*} a^{u - v} = \frac{a^u}{a^v} \text{.} \end{equation*}

In other words, the exponential of a difference is the quotient of the exponentials.

Here is another algebraic property that doesn’t have a corresponding “reflected” property in Section 8.2.

Pattern 9.5.5. Composing exponentials.

For positive base \(a\text{,}\) we have

\begin{equation*} {(a^u)}^v = a^{u v} \text{.} \end{equation*}

Justification.

From Pattern 9.3.4, we have

\begin{align*} {(a^u)}^v \amp = e^{v \ln (a^u)} \amp a^{u v} \amp = e^{(u v) \ln (a)} \text{.} \end{align*}

But from Pattern 8.2.10 we have

\begin{equation*} \ln (a^u) = u \ln (a) \text{,} \end{equation*}

so the two exponential expressions above are indeed the same.

Note 9.5.6.

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

\begin{equation*} t \to 0^+ \qquad \implies \qquad \ln (t) \to -\infty \end{equation*}

from Pattern 8.2.3 as

\begin{equation*} \text{input} \to 0^+ \qquad \text{then} \qquad \text{output} \to -\infty \text{,} \end{equation*}

then the “reflected” pattern for the exponential function is

\begin{equation*} \text{output} \to 0^+ \qquad \text{when} \qquad \text{input} \to -\infty \text{,} \end{equation*}

so that

\begin{equation*} t \to -\infty \qquad \implies \qquad \exp t \to 0^+ \text{,} \end{equation*}

meaning that the exponential graph heads towards \(0\) from above as the input goes off to the left.

Section 9.6 Exponential rate

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

\begin{equation*} A_0(t) = \ccmint{0}{t}{e^u}{u} \text{.} \end{equation*}

In the Sage cell below, try different values for the variable upper bound t and look for the pattern.

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.

In the case that the exponential rate function has a different base, we need a correction factor, similar to Pattern 7.4.2.

Pattern 9.6.1. Integral of an exponential function.

For rate function \(r(t) = a^t\) (with positive base \(a\)), we have

\begin{equation*} \ccmint{0}{t}{a^u}{u} = \frac{a^t - 1}{\ln (a)} \text{.} \end{equation*}

A quantity experiencing an exponential rate of variation can itself be modelled by an exponential function

\begin{equation*} q(t) = q_0 + \frac{a^t - 1}{\ln (a)} \text{,} \end{equation*}

where \(q_0 = q(0)\) is the initial amount.

In the case of the natural exponential function as the rate function (that is, where \(a = e\)), then since \(\ln (e) = 1\) we have

\begin{equation*} \ccmint{0}{t}{e^u}{u} = e^t - 1 \text{.} \end{equation*}

Graph demonstrating that the area under the natural exponential function is measured by the natural exponential function. — Figure 9.6.2. Area under the natural exponential function is measured by the natural exponential function.

Example 9.6.3. Using Pattern 9.6.1.

Let’s extend Example 8.1.4 by computing

\begin{equation*} \ccmint{3}{4}{ t + \frac{1}{t} + e^t }{t} \text{.} \end{equation*}

Use the same strategy as in Example 8.1.4 (and Example 7.4.6):

\begin{align*} \ccmint{3}{4}{ t + \frac{1}{t} + e^t }{t} \amp = \ccmint{3}{4}{t}{t} + \ccmint{3}{4}{\frac{1}{t}}{t} + \ccmint{3}{4}{ e^t }{t}\\ \amp = \left( \ccmint{0}{4}{t}{t} - \ccmint{0}{3}{t}{t} \right)\\ \amp \qquad {} + \left( \ccmint{1}{4}{\frac{1}{t}}{t} - \ccmint{1}{3}{\frac{1}{t}}{t} \right)\\ \amp \qquad {} + \left( \ccmint{0}{4}{e^t}{t} - \ccmint{0}{3}{e^t}{t} \right)\\ \amp = \left( \frac{4^2}{2} - \frac{3^2}{2} \right) + ( \ln (4) - \ln (3) ) + \bbrac{ (e^4 - 1) - (e^3 - 1) }\\ \amp = \frac{7}{2} + \ln \left(\frac{4}{3}\right) + e^3 (e - 1) \text{.} \end{align*}

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