Operations for functions: Exponential and logarithmic functions
Exponential functions
Exponential functions occur naturally in real life. Consider, for example, the number of bacteria that grow in a culture medium where each minute every bacterium splits into two bacteria. If we start with one bacterium, then after one minute we have two bacteria, after 2 minutes 4 bacteria, and so on:
Time (in minutes) | Number of bacteria |
0 | 1 |
1 | 2 |
2 | 4 |
3 | 8 |
4 | 16 |
5 | 32 |
If is the number of minutes that have passed and is the number of bacteria, the equation that describes the growth is Clearly, is a function of . The graph of this function is shown below.
The function is an of an example of an exponential function:
Exponential function
If is a positive real number, then the function is called the exponential function with base . Its domain is .
The reason for requiring the base to be positive is related to the fact that for negative values of the power is not always defined. For example, if and , then is a number squaring to , but no such number is real.
Later we will see that, if , the exponential function grows faster than any power function.
The function , “raising to the power ”, where the independent variable is the base , is a power function, but not an exponential function.
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