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 3: tex2html_wrap_inline155tex2html_wrap_inline159

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_inline161


eqnarray29



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


eqnarray37



Now we have an equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation tex2html_wrap_inline163


eqnarray49



Now let's look at the second factor,


eqnarray62



Now we have a second equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation tex2html_wrap_inline165


eqnarray74


eqnarray79



The exact answers are tex2html_wrap_inline167 and tex2html_wrap_inline169 The approximate answers are tex2html_wrap_inline171 and tex2html_wrap_inline173. However these answers may or may not be the solutions. You must check the answers with the original equation.



Check these answers in the original equation.



Check the solution tex2html_wrap_inline167 by substituting 0.847297860387 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 0.847297860387 for x, then x=0.847297860387 is a solution.


Check the solution tex2html_wrap_inline189 by substituting -0.287682072452 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 -0.287682072452 for x, then x=-0.287682072452 is a solution.



You can also check your answer by graphing tex2html_wrap_inline201 (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: 0.847297860387 and -0.287682072452. This means that 0.847297860387 and -0.287682072452 are the real solutions.








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