A Modeling Approach to Calculus: MA2C

Subsection 1.7.1 Properties of Exponents

Do you remember why we have powers as a mathematical notation? When the power is a positive integer, it is to represent repeated multiplication. For example, \(3^4\) means that we multiply four threes together,

How did we go from this simple notational convenience to be able to interpret negative powers or fractional powers or even irrational powers? We make sense of these more complex ideas by thinking about what properties the notation should satisfy.

We know that when we multiply and divide by the same number, the net effect is equivalent to multiplying by 1. We say that the terms cancel. We should think of these actions as inverse operations. That is, the action of multiplying a number by 3 and dividing a number by 3 are inverse operations. If you do them in succession (one after the other), the net effect is equivalent to having applied no operation at all,

Extending this idea allows us to simplify repeated factors represented by powers. How would we simplify \(\displaystyle \frac{3^5}{3^2}\text{?}\) If we realize that \(3^5\) in the numerator means that we multiply by five threes and that the \(3^2\) in the denominator means that we divide by two threes, then we recognize that there are two pairs of inverse operations:

The net effect of dividing by \(3^2\) is that we removed two of the threes in the product \(3^5\text{.}\) This is why we say that division causes powers to subtract, \(\displaystyle \frac{3^5}{3^2} = 3^{5-2} = 3^3\text{.}\)

All of the basic properties of powers are motivated by the idea that an integer power corresponds to repeated multiplication. For each property, can you think of how it would be a consequence of this idea?

We illustrate several additional properties in the context of integer powers. For integer exponents, a power means repeated multiplication (similar to how multiplication by an integer means repeated addition). So \(b^3 = b \cdot b \cdot b\text{.}\) The product properties are just about counting.

The zero power property is necessary for the power of a sum rule to remain consistent. We know that \(b^{x+0}=b^x\text{.}\) But the properties of powers also mean \(b^{x+0} = b^x \cdot b^0\text{.}\) For these to both be true requires \(b^x = b^x \cdot b^0\) or that \(b^0 = 1\text{.}\)

The properties of powers relating to products and quotients behave similar to the distributive property of multiplication over addition. This is because multiplication originates as repeated addition, just as powers originate as repeated multiplication. However, addition and powers have no convenient properties. Many mistakes occur when students forget this and imagine that powers distribute over addition like multiplication. (It doesn't!)

Subsection 1.7.2 Exponent Operations and Their Inverses

Expressions have a dual interpretation. On the one hand, an expression is a mathematical object that represents some numerical value. On the other hand, an expression also represents a sequence of operations that act on other values. For example, the expression \(3x^2+5\) is a formula that for each value of \(x\) represents a particular value. It also describes a sequence of operations: “Take a value, \(x\text{.}\) Square it. Multiply by three. Add 5.”

Thinking of powers as operations will require some caution. In the expression, \(2^3\text{,}\) the numbers \(2\) and \(3\) play different roles. The number \(2\) is called the base and the number \(3\) is called the exponent. As an action, the expression \(2^3\text{,}\) raising 2 to the power 3, can be interpreted in two ways, depending on which number we think of as being acted on. We could say that take the number 2 and apply the power 3. This is likely the more familiar interpretation. Alternatively, we could say that we take the number 3 and apply the base 2. The second interpretation corresponds to the exponential operation.

Introducing variables might help make this distinction clearer. This corresponds to thinking of the expression as the value of a function. We will let \(x\) represent the number being acted on. The operation of a power \(2^3\) corresponds to \(f(x) = x^3\) with \(x=2\text{.}\) The same expression interpreted as an exponential operation would be the formula \(g(x) = 2^x\) with \(x=3\text{.}\) An elementary power function (applying a power) raises a value to a fixed power, \(f(x)=x^p\text{,}\) for a constant \(p\text{.}\) An elementary exponential function (applying a base) uses a value as the exponent of a fixed base, \(g(x)=b^x\text{,}\) for a constant \(b\text{.}\)

Power and exponential functions have corresponding inverse operations. A root provides the inverse operation to an integer power. A logarithm provides the inverse operation to an exponential. We begin by focusing on powers and roots.

If we wish to solve the equation \(x^n = 4\) where \(n\) is a positive integer, we might graph \(y=x^n\) and \(y=4\) and look for where they intersect.

For even powers \(n\text{,}\) the equation \(x^n=4\) has two solutions. The graph is symmetric across the \(y\)-axis because the product of an even number of negative values is positive, \((-1)^2=1\text{.}\) Notice that if we were trying to solve \(x^n = -4\text{,}\) there would be no solutions when \(n\) is even but there would be a single solution when \(n\) is odd. The solution is called the \(n\)th root.

A root can be written as a fractional power. The properties of exponents imply that \((x^p)^n = x^{pn}\text{.}\) If \(p=\frac{1}{n}\text{,}\) then \((x^{\frac{1}{n}})^n = x\text{.}\) That is, \(x^{\frac{1}{n}} = \sqrt[n]{x}\text{.}\) This equivalence means that for any rational number \(p=\frac{k}{n}\text{,}\) the power \(x^p\) can be computed using integer powers (repeated multiplication) and extracting roots:

We can simplify expressions that have roots and powers applied consecutively. For example, \(\sqrt[3]{a^3}\) takes a value \(a\text{,}\) and then applies the cubing operation followed by the cube-root operation. Because these are inverses, we recover the original value,

We must be careful, again, when the inverse operations involve even powers. For example, the expression \(\sqrt{a^2}\) does not actually simplify to \(a\) in all cases. If \(a=-2\text{,}\) then squaring this gives \(a^2=4\) and then the square root gives \(\sqrt{a^2} = \sqrt{4} = 2\text{.}\) In general, when \(n\) is an even power, then \(\sqrt[n]{a^n} = |a|\text{.}\) Applying the power first always makes \(a^n\) a positive value, and then the \(n\)th root is also defined to return a positive value.

On the other hand, the expression \((\sqrt{x})^2 = x\) rather than \(|x|\text{.}\) In this situation, the first operation is the square root which requires that \(x \ge 0\text{.}\) Squaring the square root of \(x\) always recovers the value of \(x\text{.}\)

How would we define a power with an irrational value? We might approximate the irrational power. That is, we find a rational number that is close to the irrational number and use it instead. This should raise questions. Does our choice of approximation matter? How close do we need to be? Calculus helps us here by introducing the idea of limits. Limits will be central to the ideas of approximation that occur throughout calculus.

Once we know that we can raise any positive base to an arbitrary number as the exponent, we can think of an exponential as a valid operation for a given positive base. Typical graphs of \(y=b^x\) for \(b \gt 1\) and for \(0 \lt b \lt 1\) are shown below. The special case when \(b=1\) corresponds to a horizontal line and is not shown.

The graphs suggest that for any \(y \gt 0\text{,}\) the equation \(b^x = y\) will have a unique solution \(x\) for each value \(y\text{.}\) The solution is called the logarithm of \(y\) for the base \(b\text{.}\)

Notice that both roots and logarithms are defined through the equation that they solve. We can interpret them as operations that will cancel their inverse operations. That is, the consecutive operations of an exponential and a logarithm with the same base cancel one another. In a similar way, a power and its corresponding root cancel one another, although we will have to be careful with even powers because of the even symmetry.

The previous example illustrated a way that we can simplify a logarithm when the base and input are both powers of the same value. In that example, we had \(\log_4(8)\) and the base \(4\) and input \(8\) were powers of 2. We used the equivalence of the equations

and then rewrote that equation in terms of the common base \(b\) to find \(x\text{.}\)

We will learn more general techniques for simplifying logarithms later. Most scientific calculators only have logarithms for base \(b=10\) and for base \(b=e\text{.}\) The logarithm for \(b=10\) is called the common logarithm and appears on a calculator with out a base log. The logarithm for \(b=e\) is called the natural logarithm and appears on a calculator as ln. We will later prove that every logarithm can be found using one of these by the change of base formula

Subsection 1.7.3 Applications

We encountered equations that require roots and logarithms to solve when we considered exponential and power function models for data. Recall that a general power function has the form

where \(A\) and \(p\) are the model parameters. A general exponential function has the form

where \(A\) and \(b\) are the model parameters, with \(b \gt 0\) and \(b \ne 1\text{.}\)

Subsection 1.7.4 Summary

• Properties of exponents are motivated by the idea that integer powers correspond to repeated multiplication.
• Exponential functions (exponential growth or exponential decay) have the form \(f(x) = A \cdot b^x\text{,}\) with the independent variable in the exponent.
• Power functions have the form \(f(x) = A \cdot x^p\text{,}\) with the independent variable as the base of the power.
• Roots, such as the square root or cube root, are inverse operations for the power operation:
  \begin{equation*} x^n = y \qquad \Leftrightarrow \qquad x = \sqrt[n]{y}\text{.} \end{equation*}
  When \(n\) is an even integer, we require \(x \ge 0\) and \(y \ge 0\text{.}\)
• Roots can be written as reciprocal powers:
  \begin{equation*} \sqrt[n]{x} = x^{1/n}\text{.} \end{equation*}
• Logarithms are inverse operations for the exponential operation, defined for every base \(b \gt 0\) and \(b \ne 1\text{:}\)
  \begin{equation*} b^x = y \qquad \Leftrightarrow \qquad x = \log_b(y)\text{.} \end{equation*}
• The change of base rule allows us to find decimal approximations for any base using the common or natural logarithm:
  \begin{equation*} \log_b(x) = \frac{\log(x)}{\log(b)} = \frac{\ln(x)}{\ln(b)}\text{.} \end{equation*}