The exponential functions are useful for describing compound interest and growth. The exponential function is defined as:
y = m. a^{x}
where 'm' and 'a' are constants with 'x' being an independent variable and 'a' being the base.
The exponential curve rises to the right for a > 1 and m > 0 and rises to left for a < 1 and m > 0.
If x takes on only positive integral values (1,2, 3,...), y = ma^{x} is the x-th term in a Geometric Progression.
Figure
Example
Compound interest can be shown to be an exponential function. If we invest A rupees in a bank that pays r% compound annual interest then,
y_{1} = A + Ar = A (1 + r)
= amount your money will grow at the end of the first year.
y_{2} = A(1 + r) + A(1 + r)r
= A(1 + r) (1 + r)
= A(1 + r)^{2}
= amount your money will grow at the end of second year.
In general,
y_{n} = A(1 + r)^{n}
This expression is of the form y = m.a^{x} where the value of 'm' is A and the value 'a' is (1 + r). The money grows exponentially when it is paid compound interest.