If the logarithmic function is one-to-one,
its inverse exits. The inverse of a logarithmic function is
an exponential function. When you graph both the logarithmic
function and its inverse, and you also graph the line y = x,
you will note that the graphs of the logarithmic function and
the exponential function are mirror images of one another with
respect to the line y = x. If you were to fold the graph along
the line y = x and hold the paper up to a light, you would note
that the two graphs are superimposed on one another. Another
way of saying this is that a logarithmic function and its inverse
are symmetrical with respect to the line y = x.
Problem 1: Find the inverse, if it exists, to the function
If it does not exist, indicate the restricted domain where it will
exist and find the inverse subject to the restricted domain.
Solution: From the graph of f(x) you can tell the function is
one-to-one and therefore has an inverse.
We know that the composition of the original function with its inverse will take us back to x or . Therefore,
All we have to do now is solve for in the equation
The base is 6 and the exponent is x - 10.
can be written
Check your solution by graphing both functions and determining whether
they are symmetric the graph of the line y = x.
You can also check your answer by checking points. In the original equation, the domain is the set of all real numbers greater than 4, and the range is the set of all real numbers greater than 10. The domain and range of the inverse function should be just the reverse.
Choose the x = 40 and calculate f(40). = = 12. This indicates that (40, 12) is point on the graph of f(x). Let's now see if the point (12, 40) is a point on the graph of the inverse
The point (12, 40) is on the graph of the inverse.
If you would like to work another problem, click on Problem.
Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.
Author: Nancy Marcus