Arizona State University College of Liberal Arts and Sciences

Age of the Earth: Exponential and Logarithmic Functions

I. The Natural Exponential Function


When you created a "formula" for predicting the outcomes in the coin exploration, you created an exponential expression,

100/2n, n = 0,1,2,3, . . .

This can also be written as

100(1/2)n, and if we let the exponent take on values other that non-negative integers, we can change the exponent from n to x to get


100(1/2)x, where x can be any real number. Now we have created an exponential function; the number raised to the exponent x is called the base (in this case, ½). Using function notation and generalizing, we can write any exponential function in the form

f(x) = a(bx),

here a, and b are constants, b > 0, b≠0, and the independent variable x can take on any real number.


The base most used by mathematicians and scientists is the number "e". An approximate value for the number e (to three decimal places) is

e ~ 2.718,

 

but you can evaluate ex for any x on most calculators or on the Java Math Pad applet. You might try these listed below.

e2, e-5, e3.7, e-2.1

 

The exponential function with base e is called the natural exponential function,

f(x) = a(ekx),


and graphs for this function, a > 0, are shown below for k either positive or negative.

k > 0

k < 0


Note that if k > 0, the graph is rising from left to right, in this case the function is increasing and can be used to describe a growth pattern. If k < 0, then the graph is falling as you move from left to right, here the function could describe a decay pattern since it is decreasing.

For more familiarity with the natural exponential function, you could use your graphing calculator to graph the following.

f(x) = 5(ex), f(x) = e5x, f(x) = e-5x, f(x) = ex/5, f(x) = e2x, f(x) = e-x/5, f(x) = 100(e-x)

The exponential function of importance to this study is one that describes exponential decay of radioactive substances. The basic general equation for this is

N(t) = N0e-λt,

where N(t) is the number of atoms of the substance remaining after t years of its decay, N0 is the number of atoms of the substance initially (t = 0), and λ is called the decay constant (more about this later). Note the minus sign in front of the exponent, so λ > 0 in order that the function N(t) describe decay.

II. Natural Logarithmic Function

The natural exponential function in the form f(x) = ex has an "inverse" function, called the natural logarithmic function. The notation used by mathematicians and scientists for this function is

g(y) = ln(y).

 

Here the independent variable y can take on any positive real number. Its graph looks like

The reason that the natural logarithmic function is called an inverse function is because it undoes whatever the natural exponential function does. Expressed mathematically, we have

ln(y) = x if ex = y;

 

in other words,

"ln of the number y is the exponent x needed to raise e to in order to get y".

Substituting some numbers,

e3 = 20.09554 so ln(20.09554) = 3.

 

The natural logarithmic function can also be evaluated on your calculator. You can verify the above example and then try evaluating the following.

ln(5), ln(2.6), ln(0.124), ln(0.738), ln(2.718), ln(5.613)

The natural logarithmic function is quite useful in solving equations involving the natural exponential function. Here are some basic properties of ln and e that you may need to know.

1. ln(ex) = x
2. e ln(y) = y
3. ln(AB) = ln(A) + ln(B)
4. ln(A/B) = ln(A) - ln(B)
5. bx = ex ln(b)
6. ex > 0

The first property is one you might use if you are algebraically solving an exponential equation. For example, in order to solve for x in the equation

e2x = 7,

you can take the logarithm of both sides to change this to a linear equation.

ln(e2x) = ln(7)
2x = ln(7)
x = ln(7)/2

 

The fifth property tells you how to convert from one base to another. For example, we can convert from base 3 to base e.

3x = ex ln(3) = e1.0986x

 

Here's another example of changing base: let's return to the function you obtained from the coin flipping exercise,

N(f) = N0(1/2)f.

 

First, change the variable f to x,

N(x) = N0(1/2)x,

 

and then convert to base e.

(1/2)x = ex ln(1/2) = e-0.6931x,

 

so our converted function is

N(x) = N0 e-0.6931x.

Now return to the original equation and use the properties of logarithms to solve.