Chapter 4: Exponential and Logarithmic Functions

Exponential Functions

In this chapter, we study the exponential and logarithmic functions, particularly natural exponential and logarithmic functions and their properties. An exponential function is a function of the form

31

31

where the base 31 is a positive constant and the exponent 31 is the variable. Note that exponential function 31 is defined for all real numbers. Exponential functions obey following identities, which are called “laws of exponents”.

\1. 31,

\2. 31,

\3. 31

\4. 31

\5. 31,

\6. 31,

for any real numbers 31 and 31, any positive real numbers 31 and 31.

The behaviour of the graphs of the exponential functions is different for the cases 31 and 31. For 31, the shapes of graphs of all exponential functions are similar to the following graph

while for 31 the shapes of graphs of all exponential functions are similar to the following graph

31

31

Note that domain of an exponential function is 31 and range is 31. In addition, if 31 (31), then the exponential function is increasing (decreasing), respectively.

Compound Interest and the Natural Base

When an amount of money is invested in an account, which pays interest, interest is calculated for the first period (be it a day, a week, a month or a year). This interest is then added to the initial amount. Following on from that, the interest for the next period is calculated but is based on the whole figure from the first period. This interest is called compound interest.

As 31 goes to infinity, 31 approaches

31.

This number

31

is called the natural base. The exponential function

31

is called the natural exponential function. The natural exponential function satisfy “laws of exponents” as

\1. 31,

\2. 31,

\3. 31

\4. 31

\5. 31,

for any real numbers 31 and 31.

Logarithmic Functions

An exponential function 31 has an inverse function since this function is one-to-one. The inverse function of 31 is called the logarithmic function with base 31 and denoted by

31

31.

The inverse relation is

31 if and only if 31

for 31 and 31.

For 31, the shapes of graphs of all logarithmic functions are similar to the following graph

31

while for 31 the shapes of graphs of all logarithmic functions are similar to the following graph

31

The logarithmic function with base 31 satisfy following basic properties:

\1. 31,

\2. 31

\3. 31

\4. 31

where 31 and 31 are positive real numbers. In addition, we express the change-of-base formula as

31

for any positive real numbers 31.

The natural logarithmic function is the logarithmic function with base 31 (inverse function of the natural exponential function) and is denoted by

31

while the common logarithmic function is the logarithmic function with base 31 . The natural logarithmic function obeys above rules, too:

\1. 31,

\2. 31

\3. 31

\4. 31

Applications of Exponential and Logarithmic Functions

Some quantities (radioactive substance, bacteria, cells, etc.) decrease or increase at a rate proportional to their amount (exponential growth or decay). The amount of material (or bacteria, cells, etc.) at time 31 is given as

31

which is a function of time 31. The constants 31 and 31 are determined by the initial conditions, i.e. the quantities of the given material at 31. Thus, the function 31 is fully determined and we can evaluate the sought for quantities asked in the question.

We observe that 31 is positive for exponential growth problems while 31 is negative for those of exponential decay.

One of the most important applications of logarithmic models is the evaluation of magnitude of an earthquake. The magnitude of an earthquake measures the energy it releases. The intensity of an earthquake is a measure of the strength of shaking produced by the earthquake at a certain place. On the Richter scale, the magnitude 31 of an earthquake of intensity 31 is expressed by the formula

31.