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Section7.4Power Functions and Exponential Functions

Subsection7.4.1Review Properties of Exponents

When we first learned about multiplication, we thought of the process as repeatedly adding one of the factors and count with the other factor. For example, the numerical expression \(7 \cdot 4\) means to add four 7's, \begin{equation*}7 \cdot 4 \equiv 7+7+7+7 = 28,\end{equation*} while the different numerical expression \(4 \cdot 7\) means to add seven 4's, \begin{equation*}4 \cdot 7 \equiv 4+4+4+4+4+4+4=28.\end{equation*} The properties of multiplication are consequences of thinking about multiplication in such away, such as the commutative property illustrated in this example. When dealing with non-integer values, we can't really use the idea of repeated addition so different interpretations (e.g., multiplication represents an area) are required. But we insist that the rules that we found for integer values still need to be true for any number.

In a similar way, the idea of powers or exponents is based originally on the idea of repeated multiplication and then generalized to satisfy basic properties. When the power or exponent is a positive integer, we interpret this as repeated multiplication with the base multiplied by itself as many times as the power indicates: \begin{gather*} 3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81 \\ 4^3 = 4 \cdot 4 \cdot 4 = 64 \end{gather*} The properties of exponents can be illustrated using the idea of repeated multiplication.

Proof

In consequence of these properties of exponents when interpreted as repeating multiplication, we will insist on any interpretation as obeying the same rules, including negative or non-integer powers. In particular, the rule for division gives us an interpretation of negative powers as reciprocals (or multiplicative inverses) of the corresponding positive powers.

Proof

Subsection7.4.2 Power Functions and Exponential Functions

The rules we have discussed for powers provide a constructive way to compute powers, first defined for integer powers as repeated multiplication, then defined for reciprocals of integers as roots. Combining these two methods, computing a power \(p\) involving any rational number (ratio of two integers \(k\) and \(m \ge 0\), \(p=k/m\)) as a product power involving roots: \begin{equation*} a^{p/q} = (a^p)^{1/q} = \sqrt[q]{a^p}. \end{equation*} If we think of this as a function of the base where the power is held constant, this defines a family of functions called power functions.

Definition7.4.3
An elementary power function is a function where the independent variable is raised to a given rational power, \(p \in \mathbb{Q}\). \begin{equation*} \mathrm{pow}_{p}(x) = x^{p} \end{equation*} A general power function is a constant multiple of an elementary power function, \begin{equation*} f(x) = A x^{p} \end{equation*} for some constant \(A\).
Example7.4.4
  1. \(f(x) = x^2\) is an elementary power function with \(p=2\).
  2. \(f(x) = \sqrt{8x}\) is a general power function since we can rewrite \(\sqrt{8x} = \sqrt{8} \sqrt{x} = 2 \sqrt 2 \cdot x^{1/2}\). Thus \(f\) is a constant multiple of the elementary power function with \(p=1/2\).
  3. \(f(x) = \frac{3}{x}\) is a general power function since \(\frac{3}{x} = 3\cdot x^{-1}\) so that we have a constant multiple of the elementary power function with \(p=-1\).
  4. \(f(x) = x^{\pi}\) is not technically an elementary power function because the power is an irrational number. However, calculus allows us to show that it behaves in the same way as a power function in spite of having an irrational power.

On the other hand, it is also possible to consider a function where the independent variable is the power and the base is constant. These functions are exponential functions. We do not have a constructive way to find these powers for irrational values of the independent variable, but calculus (using logarithms) provides a way for us to know that the values exist and how to get arbitrarily close to those values.

Definition7.4.5
An elementary exponential function is a function where a constant positive base \(b>0\) is raised to the power of the independent variable. \begin{equation*} \exp_{b}(x) = b^{x} \end{equation*} A general exponential function is a constant multiple of an elementary exponential function, \begin{equation*} f(x) = A b^{x} \end{equation*} for some constant \(A\).
Example7.4.6
  1. \(f(x) = 2^x\) is the elementary exponential function \(\exp_2(x)\) with base \(b=2\). At integer values of \(x\), this function gives the powers of 2. \begin{equation*}f(0)=2^0=1, \: f(1)=2^1=2, \: f(2)=2^2=4, \: f(3)=2^3=8, \: f(4)=2^4=16 \end{equation*}
  2. \(f(x) = 3^{2x+1}\) is a general exponential function, although it does not yet look like it because the power is not just the variable \(x\). We have to apply the properties of powers to rewrite the formula. \begin{equation*}f(x) = 3^{2x+1} = 3^{2x} \cdot 3^{1} = (3^2)^{x} \cdot 3 = 3 \cdot 9^x\end{equation*} So \(f(x)\) is a general exponential function with base \(b=9\) and a constant multiple of \(A=3\).
  3. \(f(x) = 5^{x^2}\) is not an exponential function because the power is not just the variable \(x\) and the rules of powers do not allow it to be rewritten in an exponential form.

Subsection7.4.3Finding Models for Data

Subsubsection7.4.3.1Power Function Models

A general principle of modeling is that each model parameter typically requires one independent equation to specify that value. In the case of a power function, \begin{equation*}y=Ax^p\end{equation*} there are two model parameters, the scaling multiple \(A\) and the power \(p\). If the power is known in advance, then you only need one equation and that usually comes from a single known data point.

Example7.4.7
What is the equation of a quadratic power function passing through the point \((x,y)=(3,12)\)?
Solution

If we do not know the power in advance, then we need two equations which will necessarily come from two different data points. We will use logarithms to determine the power. Although the notation below uses the common logarithm, in fact the answer is the same regardless of whether you use the common logarithm or the natural logarithm.

Example7.4.8
What is the equation of a power function passing through the points \((x,y)=(3,12)\) and \((x,y)=(1,2)\)?
Solution

The previous example was kept simple because one data point was at \(x=1\). If that data point is not included, then the calculation is a little more complex. However, using the log-log transformation means it is really the same as finding the equation of a line and apply logarithm rules.

Example7.4.9
What is the equation of a power function passing through the points \((x,y)=(3,12)\) and \((x,y)=(6,4)\)?
Solution

Subsection7.4.3.1Exponential Models

Exponential models have a special property that a linear formula in the power can always be rewritten in an exponential form by using properties of powers. \begin{equation*} b^{ax+c} = b^{ax} \cdot b^{c} = (b^a)^{x} \cdot b^{c}\end{equation*} In this new equation, the value \(b^a\) becomes the new base and the constant \(b^c\) becomes the new scaling multiple. Consequently, we have an exponential model if we can make it almost look exponential but with a linear formula in the exponent. This is often much easier to accomplish.

Example7.4.10
Find an exponential model passing through \((x,y)=(0,4)\) and \((x,y)=(3,1)\).
Solution

The previous example was easy because \(x=0\) was given for one of the data points. If this is not provided, then the calculation is a little trickier.

Example7.4.11
Find an exponential model passing through \((x,y)=(2,4)\) and \((x,y)=(6,12)\).
Solution