Inverse Function of Exponential Function
Activation of Prior Knowledge
We have already defined the rational and irrational exponent of any positive real number of a. Stating this, in case a is a positive real number and a ≠ 1, we call the function of f : IR → IR, f(x) =
as exponential function.
The domain of the function f(x) = is real numbers. We have already come up with different exponential functions by changing a.
For instance, the functions , or are exponential functions.
Let’s now find out the values of the function for the x values x= -4, -3, -1, 0, 1, 2 and remember how we could draw its graph.
as exponential function.
The domain of the function f(x) = is real numbers. We have already come up with different exponential functions by changing a.
For instance, the functions , or are exponential functions.
Let’s now find out the values of the function for the x values x= -4, -3, -1, 0, 1, 2 and remember how we could draw its graph.
f(x)=graph of 3^x
You state
How is the graph of
The Definition of Logarithmic Function
When a and b are positive real numbers and a ≠ 1, we have already defined the number of (the logarithm of b in relation to the base of a)
Let’s now remember: It is equal to c and the exponent of is equal to b. Saying this, we have the value of ……….. . We have the function ………… when we replace b with x in the definition of the logarithm of b in relation to the base of a. We call this function as logarithmic function with a base.
The domain of logarithmic function is all positive real numbers. The logarithmic function could also be defined as the inverse function of exponential functions. We have already stated that exponential functions are bijective. Depending on this, if we find x, we will have. As we have explained in the definition of a inverse function, if we replace x with y and y with x, we will end up with the logarithmic function, which is
the inverse function of y. Now, let’s examine the graph, which is the inverse function, for the values of a>1 , 0<a<1 and a<0 by comparing them.
Question 1) How is the graph for a<0? What is your opinion about the possible
reason behind it? (Clue: What is the range for exponential functions?)
Question 2) How does the graphic change for 0<a<1 and a>1?
Question 3) What could you say about the increase and decrease in the logarithmic function? For which values of a is it increasing and decreasing?
Let’s now observe the changes in the function for the values a, b, and c.
Task: Try to draw the graphs of and with the help of your tablet PCs.