On-line Math 21

3.2  Exponential and Logarithmic functions

3.2.1  Basic definitions

Exponential functions are functions of the sort f(x) = a^x . a is called the base of the exponential., and has to be positive. They are distinctly different from power functions, although students often confuse them. For example, if a = 2 , the function f(x) satisfies f(0) = 1 , f(1) = 2 , f(2) = 2 , f(3) = 8 , f(-2) = 1/4 . Here is a great plot of various exponential functions (various bases, that is).

Rules

There are only two basic algebraic rules for exponentiation. They are, when you think about them in terms of integer powers, obvious.

Proposition 1 For all numbers a, b, and c ,

abac = ab+c, and ( ab) c = abc.

The primary reason to study exponential functions is that they model many natural phenomena, that is, the behavior of many systems are expressed in terms of exponential functions. Such things as compound interest, radioactive decay, and population growth are governed by exponential functions.

One curious number always mentioned in this context is e . e is just a number, like p, which has a specific value. That letter, e , is almost-always reserved for that particular number (again, like p » 3.14159 . e is approximately 2.71828 . But e, like p, comes from a specific relationship. If you look at the graphs of the various exponential functions y = a^x for a number of choices of a , they all pass through (1,0) , since a⁰ = 1 for any positive number a . e is that choice of base so that the slope of the curve as it passes through (1,0) is 1.

By the way, just like p, e is irrational.

Logarithms

Be sure to read throught the general material about inverse functions.

Logarithms are the inverse of exponentiation, that is, log_ax is the inverse of f(x) = a^x . Said another way, logarithms are just exponents in reverse:

y = logbx Û  x = by.

So, log₄2 = 1/2 , log₃(1/3) = -1 , and so on. There is a special case, when the base is e . Then, instead of writing log_ex , we write lnx , the natural logarithm of x . There is also another notation for log₁₀x . It is sometimes called logx , the common logarithm of x . Unfortunately, most post-calculus math texts now confuse the notational issue even more, by writing the natural logarithm of x as logx . Hopefully, it will be clear from the context which is meant. To a mathematician, there is no special significance of the number 10, except for the number of fingers we have.

Rules for Logarithms:

These are just rules for exponents, backwards. In fact, the logarithm functions are precisely the inverse of exponentials:

logb(bx) = x and blogb(x) = x,

for all bases b and all values of x for which the expressions make sense. In particular, ln(e^x) = x , and e^lnx = x .

Proposition 2 For all b , c , and d , c and d > 0 ,

1. log_b(cd) = log_b(c)+log_b(d) ,
2. log_b(c^d) = dlog_b(c) , Here d can be negative.
3. 
   

   logb(c) logb(d) = logd(c).

   This last fact means that you can find the logarithm of c to any base, if you know all logarithms of one base.

Proof:

Here are some equations involving logarithms:

Example 1 log₂80-log₂5 = log₂(80/5) = log₂16 = 4 .

Example 2 2^x = e^x ln2 .

Solution

Exercise 1 Solve for x : lnx+ln(x-1) = 0 .

x =

The graphs of the various logarithm functions, for different bases b , are shown below. The bases used are b = 2 , e , 3 , and 10.

Email Address (Required to submit answers):