## SOLVING EXPONENTIAL EQUATIONS

Note:

• To solve an exponential equation, isolate the exponential term, take the logarithm of both sides and solve.

If you would like an in-depth review of exponents, the rules of exponents, exponential functions and exponential equations, click on exponential function.

Solve for x in the following equation.

Example 4:

In order to solve this equation, we have to isolate the exponential term. Since we cannot easily do this in the equation's present form, let's tinker with the equation until we have it in a form we can solve.

We cannot easily factor this problem. Therefore, let's see if we can use the Quadratic Formula to solve the problem even thought the equation does not look like a quadratic equation. In fact, it is a quadratic equation in

Let's rewrite the equation with the following substitutions: and

Now you should recognize this as a quadratic equation in p where a=1, b=6, and c=-15.

We have two answers for p. However, the original equation did not contain a p. We used p to put the original equation into an equation we could solve. Now, take the p back to

Take the natural logarithm of both sides of the equation

The exact solution is and the approximate solution is This answer may or may not be the solution. You must check it in the original equation.

Check the answers in the original equation.

Check the solution by substituting 0.641316629016 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 0.641316629016 for x, then x=0.641316629016 is a solution.

You can also check your answer by graphing f(x)= (formed by subtracting the right side of the original equation from the left side). Look to see where the graph crosses the x-axis; that will be the real solution. Note that the graph crosses the x-axis at one place: 0.641316629016. This means that 0.641316629016 is the real solution.

If you would like to work another example, click on Example

If you would like to test yourself by working some problems similar to this example, click on Problem

If you would like to go to the next section, click on Next

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Author: Nancy Marcus