SOLVING EXPONENTIAL EQUATIONS


Note:

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 1:tex2html_wrap_inline155tex2html_wrap_inline114


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 tex2html_wrap_inline116


eqnarray30



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


eqnarray38



Now we have an equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation tex2html_wrap_inline118


eqnarray45



Now let's look at the second factor,


eqnarray50



Now we have a second equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation tex2html_wrap_inline120


eqnarray57



The exact answers are x=0 and tex2html_wrap_inline124



Check these answers in the original equation.



Check the solution x=0 by substituting 0 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.


Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 0 for x, then x=0 is a solution.


Check the solution tex2html_wrap_inline140 by substituting 0.69314718056 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.


Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 0.69314718056 for x, then x=0.69314718056 is a solution.



You can also check your answer by graphing tex2html_wrap_inline152 (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 two places: 0 and 0.69314718056. This means that 0 and 0.69314718056 are the real solutions.








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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