The exponential and logarithmic functions

The exponential function

y=exp(x)

is very important in mathematics. As is the natural logarithmic function

y=log(x)

In this lesson it is explained how these functions are defined, their properties are studied and the close relationship that exists between them is shown. 

The exponential function.

To start with, we consider the function 2^x. The pupil is most probably familiar with the values of this function (2, 4, 8, 16, 32, 64, ...) for whole values of  x. Obviously it is a question of a function that grows a lot on increasing the value of x. For whole negative numbers of x, the values of the function are also very well known (1/2, 1/4, 1/8, 1/16, ...). However, what does 2^x mean when x is not a whole number? The general definition is made for rational numbers m/n with m and n as whole numbers, and then it is extended by continuity to all real numbers. 2^(m/n) is defined as the n-th root of 2^m. This definition makes 2^x a very difficult function to evaluate for values of x which are not whole numbers or very simple fractions. The following applet shows the graph of that function and the pupil can explore its values by modifying the value of x. In the same applet you can study the exponential functions a^x for different values of a.

The derivative of the exponential functions can be calculated as the limit when h tends towards zero of 

(a^(x+h)-a^x)/h

which is the same as a^x multiplied by the limit when h tends towards zero of

(a^h-1)/h

This apparently simple limit is difficult to calculate.   The following applet allows us to estimate its value basing it on the values h=1/n for large values of n. The graph that appears is that of the function (a^x-1)/x.

It is suggested to the pupil to find the number a for which the limit is equal to 1.0. This number is the famous number e that we will find many times in this lesson. 

The following applet shows the function a^x for several values of a and also an approximation of its derivative (using h=0.000001). The pupil can check that when a=2.72818 the function a^x and its derivative appear to coincide.

In reality a number exists, the famous number e, for which the function e^x is equal to its derivative. The exponential function is defined using this number, namely: exp(x)=e^x. The value of e has been calculated with many decimals.  This is the value of e up to the twentieth decimal place: 

e=2.71828182845904523532874

In the majority of calculations only the first five decimal places are used, that is to say

e=2.71828

We see an intuitive way of calculating e based on the fact that the derivative of e^x should be equal to e^x.  In order tha the limit when h tends towards zero of (e^(x+h)-e^x)/h be equal to e^x, it is necessary and sufficient that the limit when h tends to zero of (e^h-1)/h is equal to 1. Intuitively it is seen therefore that for small values of h, e is more or less equal to (1+h)^(1/h). Changing h for 1/n with n as being whole number, we have that for large values of n e is aproximately equal to(1+1/n)^n. In other words, e is equal to the limit, when n tends to infinity, of(1+1/n)^n. The following applet allows good estimations of the number e to be obtained as the limit of (1+1/n)^n when n tends towards infinity.

In the study of the logarithmic function we will see another way of arriving at the number e.

The logarithmic function.

The logarithmic function can be defined as the inverse function of the exponential. That is to say: log(x)=y if x=exp(y). The following applet shows the graphs of the exponential and logarithmic functions and shows the relationship that there is between them both.   

The graph of the logarithmic function is the reflection of the graph of the exponential with repect to the straight line y=x

A logarithmic function can be defined for each exponential function a^x. To distinguish one from another, the inverse function of a^x is called the logarithmic function with base a. The logarithmic function with base e is also called the natural logarithm. The following applet shows all the exponential functions a^x and the corresponding logarithmic functions with base a, which we denote by loga(x).

It is easy to see that loga(x)=log(x)/log(a). Indeed, a=e^(log(a)) and therefore, using the properties of the exponentiation:

a^(log(x)/log(a))=e^(log(a)*(log(x)/log(a))=e^log(x)=x.

The following example illustrates the reason why logarithms had great importance in the past to simplify numeric calculations.  The reason is that by using logarithms, a multiplication can be converted into a simple sum.   The applet shows two values x1 and x2, their corresponding logarithms y1 and y2, and also the product x1*x2 and the sum y1+y2. The pupil will see that x1*x2 = y1+y2 always.

For this reason, in times before computers, tables of logarithms were constructed.  If they wanted to multiply the numbers x1 and x2, they found in the table their logarithms y1 and y2, and added them obtaining a value y and they found the value of x in the table for which log(x)=y. Therefore x is equal to the product x1*x2. Nowadays this method is unnecessary, it only has historic importance.  However the exponential function continues to be important in science for other reasons that are more related to the first property we studied, that of it being equal to its derivative. This fact means that the exponential function appears continually in the description of physics phenomena like for example, radioactive decay.        

The following example shows the behaviour of radioactive material over time.

Finally, we will study the logarithmic function independently of the exponential.  We assume that there exists a function f(x) that has the property that

f(x*y)=f(x)+f(y)

for all positive values of x and y. It is easy to see that this one property implies that:

f(1)=0 and f(x^n)=n*f(x)

Therefore, writing the expression to estimate the derivative with an increment h=x/n, we obtain:

(f(x+x/n)-f(x))/(x/n)=(1/x)f((1+1/n)^n)

Taking the limit when n tends to infinity, we obtain that the derivative of f´(x) is equal to f(e)/x, where

e=lim (1+1/n)^n
n->infinity

To summarize:

f´(x)=f(e)/x

This result tells us that the functions the satisfy f(x*y)=f(x)+f(x) all have the same derivative, except for a multiplying factor.  It is natural therefore to define the logarithmic function as the integral (from 1 to x) of the function 1/x and therefore it results that 

log´(x)=1/x

and hence

log(e)=1

For this reason it is said that e is the base of the natural logarithms, also called Neperian. The last applet shows the graphs of the functions log(x) and 1/x and the pupil can check that the estimation of the derivative (log(x+h)-log(x))/h is almost equal to 1/x for the value of h=0.000001

Author: José Luis Abreu León

	Ministerio de Educación, Cultura y Deporte. Año 2000