Exponential Functions

Earlier we went through exponentials expression where it was explained that Exponential Expressions are just a way to write repeated multiplication of the same number in short form.

Base and Exponents:

• So in the case of 32, it can be written as 2 × 2 × 2 × 2 × 2 = 2⁵

• The Exponent indicates the number of times the base is used as a factor. Exponent = 5

• The Base and the number that is multiplied by itself repeatedly (the factor). Base 2

2⁵ We read this expression as “two to the fifth power”.

In general, we will have that aⁿ = a × a × a... × a, (n times), where a will be the base and n the exponent, and we will read it as “a to the nth power”.

The exponential expression a^x = a × a × a... × a, (x times), where a will be the base and x is a variable exponent, is read as “a to the x power”.

NOTE: a can be any positive number ( a > 0).

Since the value of x varies, we can determine the instantaneous value of the expression by expressing it as a function by introducing a dependent variable, say y.

y = a^x

So, exponential functions are in fact exponential expressions with variable exponents

• When a=1, the graph is a horizontal line at y=1
• Apart from that there are two cases to look at:
  1. a between 0 and 1
  2. a above 1

• 'a' is always greater than 0, and never crosses the x-axis
• 'a' is always intersects the y-axis at y=1 ... in other words it passes through (0,1)
• At x=1, f(x)=a ... in other words it passes through (1,a)
• It is an Injective (one-to-one) function

Exponential decay typically is represented by 'Strictly Decreasing' exponential function graphs.

In exponential decay, a quantity slowly decreases in the beginning and then decreases rapidly. We use the exponential decay formula to find population decay (depreciation) and we can also use the exponential decay formula to find half-life (the amount of time for the population to become half of its size). Let us learn more about the exponential decay formula along with the solved examples