Introduction
Definition

A function f of the form f (x) = a^x, a > 0 and a ≠ 0 is called an exponential function with base a.

Note

• Domain of f (x) = a^x, a > 0 is the set of all real numbers.
• Range of f (x) = a^x, a > 0 and a ≠ 1 is the set of positive real numbers and if a = 1, then the range is {1}.

The exponential function y = a^x

Let us see the behaviors of y = a^x, a > 0, and its graph.

In the exponential function y = a^x, a > 0.

1. The y-intercept is a⁰ = 1 and the graph of y = a^x cross the y-axis at (0,1).

2. No x-intercept (the graph will not cross the x-axis).
The graph of y = a^x is completely above the x-axis.
The x-axis or the line y = 0 is a horizontal asymptote.
y = a^x will never be 0.

(a) For 0 < a < 1, the graph approaches the positive x-axis as x increases to very large number and far up for large negative x.
(b) For a > 1, the graph approaches the negative x-axis as x decreases and the graph will be far up as x increases.

3. For a > 1
(a) If x > 0, then a^x > 1.
(b) If x < 0, then 0 < a^x < 1.

4. For 0 < a < b < 1
(a) If x < 0, then a^x > b^x.
(b) If x > 0, then a^x < b^x.

5. y = a^x is a one-to-one function

Exponential equations

In exponential equations there is a variable exponent. In solving exponential equations, we will use the exponential properties and in solving such equations it is important to note that if a^x = 1, then x = 0. It is due to a⁰ = 1, where a ≠ 0.