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.


Problem 7.3c:

tex2html_wrap_inline220


Answer:The exact answers are tex2html_wrap_inline222 The approximate answers are tex2html_wrap_inline224 and 1.89818568407.

There are many forms of the exact answer; however, all forms will have the same approximate answer.



Solution:


Isolate the exponential term in the equation tex2html_wrap_inline228 .


Divide both sides of the equation by 2


eqnarray43


eqnarray48






Take the natural logarithm of both sides of the equation tex2html_wrap_inline232


eqnarray64


eqnarray69


eqnarray77





Solve for x using the quadratic formula tex2html_wrap_inline234 where tex2html_wrap_inline236


eqnarray91


eqnarray97


eqnarray102


eqnarray107


eqnarray112


The exact answers are tex2html_wrap_inline238 and the approximate answers are tex2html_wrap_inline224 and 1.89818568407.

tex2html_wrap_inline244


Check these answers in the original equation.





Check the solution tex2html_wrap_inline248 by substituting 1.89818568407 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 1.89818568407 for x, then x=1.89818568407 is a solution.




Check the solution tex2html_wrap_inline260 by substituting 0.60181431593 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 0.60181431593 for x, then x=0.60181431593 is a solution.




You can also check your answer by graphing tex2html_wrap_inline272 (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 0.60181431593 and 1.89818568407. This means that 0.60181431593 and 1.89818568407 are the real solutions.




If you would to review the answer and solution to problem 7.3d, click on Solution.


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