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 2:

tex2html_wrap_inline278


The first step is to isolate the expression tex2html_wrap_inline280

Subtract 14 from both sides of the equation.


tex2html_wrap_inline284


tex2html_wrap_inline286


Divide both sides of the equation by 12.


tex2html_wrap_inline290


tex2html_wrap_inline292





Take the natural logarithm of both sides of the equation tex2html_wrap_inline294


tex2html_wrap_inline292


tex2html_wrap_inline298


tex2html_wrap_inline300


tex2html_wrap_inline302


tex2html_wrap_inline304





Use the Quadratic Formula tex2html_wrap_inline306 where a=1, b=-9, tex2html_wrap_inline312 .


tex2html_wrap_inline314


tex2html_wrap_inline316


tex2html_wrap_inline318


tex2html_wrap_inline320 tex2html_wrap_inline322


tex2html_wrap_inline324 tex2html_wrap_inline326




The exact answers are tex2html_wrap_inline318 and the approximate answers are tex2html_wrap_inline330 and tex2html_wrap_inline332




Check these answers in the original equation.




Check the solution tex2html_wrap_inline320 by substituting tex2html_wrap_inline322 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 8.05483513081 for x, then x=8.05483513081 is a solution.




Check the solution tex2html_wrap_inline348 by substituting tex2html_wrap_inline326 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 0.94516486919 for x, then x=0.94516486919 is a solution.





You can also check your answer by graphing tex2html_wrap_inline360 (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 8.05483513081 and 0.94516486919. This means that 8.05483513081 and 0.94516486919 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.

If you would like to go back to the equation table of contents, click on contents.


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