## 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 2:

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.

Factor the left side of the equation

The only way that a product can equal zero is if at least one of the factors is zero.

This is impossible. Therefore, If you did not make this observation, you would have caught it when you tried to take the natural logarithm of both sides since you cannot take the logarithm of a negative number.

Now let's look at the second factor,

Now we have a second equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation

Check this answer in the original equation.

Check the solution by substituting 1.09861228867 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 1.09861228867 for x, then x=1.09861228867 is a solution.

You can also check your answer by graphing (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: 1.09861228867. This means that 1.09861228867 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

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Author: Nancy Marcus