Are you tired of adding too many numbers with the same value? Do you find it boring and monotonous? Are you sick of that long method which consumes most of your time? Well, mathematics has ways of dealing with such problems in a short and simple manner.
What does it mean to raise a number to the power of x?
The operation of raising a number to a power is a special case of multiplication in which the factors (or, the numbers to be multiplied) are all equal.
[Note: In the succeeding texts, remember that the multiplication operation is denoted by an “x”; However, when used to denote power or exponent, the “x” is italicized; and these are two different things.]
For instance, in the following examples,
For instance, in the following examples,
So, the expression 53 (read as: “five raised to the power of 3” or “five to the third power” or “five cubed”) means that three 5’s are to be multiplied successively, meaning 5 x 5 x 5. Similarly, 42 (read as: “four raised to the power of 2” or “four raised to the second power” or “four squared”) means 4 x 4.
Anyway, those things are basic.
Anyway, those things are basic.
The previous statements are information about raising a number to a certain power. However, raising a number to x is a special case. In the language of mathematics, this is called an EXPONENTIAL FUNCTION and can be written as:
with the base a which could be any positive real number. I mean, exponential functions always have some positive number other than 1 as the base. It has many implications, however, this function is easy to evaluate. A positive exponent means that the function is increasing while a negative exponent makes the said function decrease.
with the base a which could be any positive real number. I mean, exponential functions always have some positive number other than 1 as the base. It has many implications, however, this function is easy to evaluate. A positive exponent means that the function is increasing while a negative exponent makes the said function decrease.
Given the exact relationship, when x is increased by 1 over what it had been, f(x) is increased to twice of what it had been.
One of the most used examples of exponential function is the exponential growth. When somebody says that the population growth is doubled every year, they are talking about exponential growth. We end at this conclusion because we already know the fact that there is a consistent fixed time interval during which the function will double, triple and so on.
Perhaps, you’ve heard of the expression “starting slow, but then growing faster and faster all the time". That's exponential function!
