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Exponential and Logarithmic Functions

Now that electronic calculators are widely available, logarithms may not get proper attention in algebra classes. Logarithmic bases other than 10 never did get that much attention and neither did the natural base. Therefore it seems wisest to treat the subjects de novo from the perspective of the calculus.

Although historically logarithms came first, owing the urgent need for ways to facilitate calculations for navigation, exponential functions are the logical beginning in the differential calculus.

Exponential Functions

The characteristic and defining feature of exponential functions is that the variable (or at least the variable under consideration) is in the exponent.

Examples of exponential functions:

{$$ \begin{align} y &= e^x \cr f(x) &= 2^x \cr g(y) &= 10^y \end{align} $$}

The part that is raised to the variable power (that has a variable exponent) is called a base.

Negative Bases Excluded

Generally, functions with negative bases are excluded from consideration. The reasons for this are the odd powers of negative numbers are negative, and even powers of negative numbers are even; and odd roots of negative number exist, whereas even roots of negative numbers do not exist (in the real numbers), making fractional exponents very dicey. While

{$$ y = (-2)^x $$}

does exist for selected values of {$ x $}, clearly it is not continuous even in small regions.

It is important not to confuse exponential functions with non-exponential functions involving exponents:

Examples of non-exponential functions:

{$$ \begin{align} y &= x^5 \cr f(x) &= x^n \cr g(y) &= y^{10} \cr v &= 5^\pi \end{align} $$}

The example above with {$f(x)$} illustrates the usefulness of the {$f()$} type of notation. Although {$n$} may be unknown or unspecified, the variable of the function is {$x$}.

The careless error to be avoided here is the attempt to use the powers rule on an exponential function.

{$$ {d \over {dx}} v^x \color{red}{\ne} xv^{x-1} \color{red}{\text{ (do not use power rule)}} $$}

Graphs of Exponential Functions

Generally, the exponential function which gives rise to the common-language "exponential" in expressions like "exponential growth" (often used incorrectly), is considered when the base is greater than 1.

Such a function is very nearly 0 for values of x somewhat less than 0 (the "somewhat" depends on the base). The function rises to the reciprocal of the base at x = -1, to 1 at x = 0 and to the value of the base at x = 1, and then increase relatively rapidly without bound (the "relatively" depends on the base). Graphs of population growth have made this shape familiar.

When the base is a fraction less than 1, the graph plunges from unbound heights on the right (negative values of x) to hit the reciprocal of the base at x = -1, 1 at x = 0, and the base at x = 1. For curve sketching, these three points are usually easy places to start.

The relationship being described here is easier to understand when it is remembered that a negative exponent means the the reciprocal of the base raised to the positive value of the exponent:

{$$ \begin{align} b^{-x} &= \left( \frac{1}{b^x} \right) \cr b^x &= \left( \frac{1}{b^{-x}}\right) \end{align}$$}

Since negating x amounts to flipping the x-axis, an exponential function with a fractional base is the mirror image of the function with the reciprocal base (or with exponent negated). Figure 1 shows these relationships, which were confusing me until I got the figure.


Figure 1: Inverting the base has the same effect as negating the exponent.


Figure 1a: As the base gets nearer 1, the exponential flattens.

Since 1 to any power is 1, exponential functions with bases near one are nearly flat, while exponential functions with base exactly 1 are the very flat y = 1.

Logarithm Functions

To make a long story short, logarithm functions are the inverses of exponential functions. As such, their graphs are the reflections about the line y=x of corresponding exponential functions.


Figure 2 illustrates the symmetry of an exponential function and the logarithm function with the same base. Functions of other bases have this same property, but the fact that the slope of the exponential function is 1 at x=0 is a defining property of e. (But we have not yet shown e exists or how its value might be found.)

Definition of Logarithm

{$$ \begin{gather} \text{for } b \ge 0\text{, } b \ne 1 \text{, } \text{ and } y \gt 0 \text{, } \cr y=b^x \iff x = \log_b y \end{gather} $$}

Negative b is excluded for reasons already discussed. b = 1 is excluded because then all powers of b would be 1, the only number that would have a log base 1 would 1, and log base 1 of 1 would be anything. And y > 0 saves me a lot of confusion. Actually, it is really just a consequence of b being non-negative. As such it probably does not need to be stated for those using the exponential form, but it is helpful reminder not to attempt to take the logarithm of a negative number.

A pair of immediate consequences of this definition are:

{$$ \begin{align} \log_b b^x &= x \cr b^{\log_b x} &= x \end{align} $$}

Then

{$$ \begin{align} \text{Let } f(x) &= b^x \cr \text{and } g(x) &= \log_b x \text{,}\cr f \left( g(x) \right) &= f \left( \log_b x \right) = b^{\log_b x} = x \cr g \left( f(x) \right) &= g \left( b^x \right) = log_b b^x = x \end{align} $$}

or in other words, exponentiation undoes what logarithm does, and logarithm undoes what exponentiatoin does, and thus the functions are truly inverses. Thanks, Captain Obvious, for restating the definition!

However, this probably a good place to review the laws of exponents and their implications for exponential and logarithmic expression.


Figure 3: logs with various bases. Inverses of some exponential functions in Figure 1a.


Figure 4: logs and exponentials with fractional bases.

While good for exercises, bases other than 10 and e seldom show up in real life. Now that electronic calculators are ubiquitous, there is not even much use for base 10. In any event, log when written without the base, means log base 10, and ln means base e.

Now e is just a constant, like π. It is written e because it is an irrational number (near 2.71). It has many interesting properties and crops up in unexpected places. For some of these properties it is considered the "natural" base for logarithms.

The Derivatives

The key results are:

{$$ \begin{align} {d \over {d/x}} e^x &= e^x , \text{ and} \cr {d \over {d/x}} \log_a x &= {1 \over x}log_a e, \text{ with the special case} \cr {d \over {d/x}} \ln x &= {1 \over x} \end{align}$$}

While there is no reason to doubt any of this, formalizing it at this level (or perhaps I am just stupid) seems very difficult. Here is a start.


Sources:

  1. FooPlot: Online graphing calculator and function plotter
  2. File:Leonhard Euler by Handmann .png - Wikimedia Commons
  3. File:John Napier (Neper).jpg - Wikimedia Commons

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Categories: Algebra, PreCalculus

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This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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