Exponential, Logarithmic, and Inverse Functions in Edwards & Penney

presentation of Edwards & Penney's work, by Corry Shores
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Exponential functions take the form

where a > 0. The exponent x is the variable here; the number a, called the base, is constant. Thus:

An exponential function

is a constant raised to a variable power, whereas the power function

is a variable raised to a constant power.

In elementary algebra, a rational power of the positive real number a is defined in terms of integral roots and powers. If n is a positive integer, then

and

If r = p/q where p and q are integers (with q positive), then the rational power

is defined by

The following is a list of the laws of exponents for all rational exponents r and s:

and also:

for any positive real number a.

Derivatives of Exponential Functions:

To compute the derivative of the exponential function

we begin with the definition of the derivative and then use the first law of exponents

to simplify it. Hence

when we substitute a^x for f (x). Then we use the laws of exponents given above to obtain

and then because a^x is "constant" with respect to h , we then pull it out of the equation to obtain

Then, under the assumption that

is differentiable, it follows that the limit

exists.

Although its value m(a) depends on a, the limit is a constant as far as x is concerned. Thus we find that the derivative of a^x is a constant multiple of a^x itself:

Because a^0 = 1, we see from the above equation that the constant m(a) is the slope of the line tangent to the curve

at point (0, 1), where x = 0.

There numerica data in this chart below

suggests that

and

The tangent lines with these slopes are shown below:

Thus it appears that

We want to avoid such awkward numerical factors as these above. It seems plausible that the value m (a) defined in this equation

is a continuous function of a. If so, then because m (2) <>and m (3) > 1, the intermediate value theorem implies that m (e) = 1 (exactly) for some number e between 2 and 3. If we use this particular number e as the base, then it follows from this equation

that the derivative of the resulting exponential function

So in other words, the function e^x is its own derivative. [To review: we looked at a function with a base whose power is x. We then added it to the differential equation, performed algebra on it, and found that

But when we differentiate on some point a, the derivative is always the slope times the function point. We know that in this function a^0 = 1 (that is, where the line is at the zero point of x, then the y is at the point 1, see figure 7.1.8 above). So we are looking for an a value that when taken to the proper power for that a value, this new value times the slope equals that powered value. So again, we want an a value for the function a^x which when differentiated gives that same value a^x. This is conceivable, because when we vary the values for a from 2 to 3, it swings past 1. If it were one, then its differential value is equal to it, becuase then the differential value equals the differentiated slope at value a times value a^x.]

We call

the natural exponential function. We see its graph below, as well as some of its estimates:

We later find that e is given by the limit

e is irrational, and estimated to 15 places is

Interlude: Logarithmic Functions and Inverse Functions:

Logarithms are "inverse" to exponential functions. The base a logarithm of the positive number x is the power to which a must be raised to get x. In other words:

Inverse Functions: The two functions f and g are inverse functions, or are inverses of each other, provided that 1) The range of values of each function is the domain of definition of the other, and the relations

hold for all x in the domains of g and f, respectively.

In other words, if we combine the functions in either order, and they produce the same result, they are inverse functions:

Thus for logarithms:

Differentiation of an Inverse Function:

Suppose that the differentiable function f is defined on the open interval I and that f ' (x) > 0 for all x in I. Then f has an inverse function g, the function g is differentiable, and

Return to the Natural Logarithm:

The natural exponential function

is defined for all x and

If f is the inverse function that consequently is guaranteed by the Differentiation of an Inverse Function theorem given above, then

Thus g(x) is "the power to which e must be raised to get x," and therefore is simply the logarithm function with base

The function g is therefore called the natural logarithm function. It is commonly denoted by the special symbol in:

Because

for all x, it follows that ln x is defined only for x > 0. The graph of y = ln x is shown below:

We see that it rises slowly when x is large. We note that ln 1 = 0, so the graph has x-intercept x = 1, and that ln e = 1 (because

The inverse function relations between

and g(x) = ln x are these:

Derivatives of Logarithmic Functions:

To differentiate the natural logarithm function, we can apply the Differentiation of an Inverse Function theorem:

and thereby write:

Or, we could begin with equation

and differentiate both sides with respect to x, in the following way (using the chain rule not provided above):

Thus we see that either way, the derivative

of the natural logarithm function is given by

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.39, 428-436.