# LOGARITHMS AND THEIR INVERSES

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.

Example 5: Find the inverse of

Comments: Recall that the composition of a function with its inverse will take you back to where you started. For example, suppose the rule f(x) will take a 3 and link it to 10; then the rule will take the 10 and link it back to the 3. Another way of stating this is . A general way to stating this is for any x in the domain of f(x).

Solution: With respect to this problem,

The base is e, the exponent is x, and the problem can be converted to the exponential function

If you graph the problem, notice that the graph is one-to-one. Therefore, we not have to restrict the domain. Notice that the domain is the set of all real numbers .

Step 1: Solve for in the equation

Step 2: Square both sides:

Step 3: Add 7 to both sides of the above equations:

Step 4: Divide both sides by 3:

If you would like to work a problem, click on Problem.

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