An Exponential Function is a function which involves exponent which is the variable part rather than the base as in any normal function. For instance f(x)= x^3 is a function and an exponential function is something like g(x)= 3^x, here the exponent or the power is a variable (x) and the fixed value is the base (3). So, the definition of Exponential functions can be given as a function whose base is a fixed value and the exponent a variable. Example: f(x) = 5^x, here the base 5 is fixed value and the exponent ‘x’ is the variable.
In general, we can define Exponential Functions as a function which is written in the form ‘a^x’ in which ‘a’ is the base which is a fixed value or constant (‘a ‘not equal to 1) and ‘x’ the variable which is any real number. The most common exponential function we come across in math is e^x which is known as the Euler’s number.
Let us now take a quick look at the Exponential Function Properties. Consider the Exponential function f(x) = b^x for which the properties are as follows:
• The domain of the exponential function consist of all real numbers
• The range is the collection of all positive real numbers
• When b is greater than 1 then the function is an increasing function also called exponential growth function and when b is less than 1 then the function is a decreasing function also called exponential decay function
• The other properties that an exponential function satisfy are,
1. b^x.b^y = b^(x+y) [when bases are same and a multiplication operation then we can add the powers]
2. b^x/b^y = b^(x-y)[when bases are same and a division operation then we can subtract the powers]
3. (b^x)^y = b^(xy) [when a base is raised to a power x and raised to whole power y then we can multiply the powers]
4. a^x.b^x= (a.b)^x [when bases are different with the same power and a multiplication operation then we can multiply the bases whole raised to power]
We come across a function called an Inverse Exponential Function; this is nothing but a logarithm function. We know that the exponential function is written in the form f(x) = b^x, to find the inverse of a given function we need to interchange x and y and solve for y. By interchanging we get x = b^y and then solving for y gives us y = log x (base b) which is a logarithm function.
