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 7.3a:

tex2html_wrap_inline221



Answer:The exact answer is tex2html_wrap_inline223 or tex2html_wrap_inline225 The approximate answer is tex2html_wrap_inline227 .


Solution:


Isolate the exponential term.


Divide both sides of the equation by 5


eqnarray41


eqnarray44





Take the natural logarithm of both sides of the equation tex2html_wrap_inline231


eqnarray58


eqnarray63


eqnarray70


eqnarray72


eqnarray76


The exact answer is tex2html_wrap_inline233 and the approximate answer is tex2html_wrap_inline235





When solving the above problem, you could have used any logarithm. For example, let's solve it using the logarithmic with base 5.

eqnarray88


eqnarray93


eqnarray104


eqnarray116


eqnarray125


eqnarray134


eqnarray143


eqnarray155


Check this answer in the original equation.





Check the solution tex2html_wrap_inline241 (can also be written in the equivalent form tex2html_wrap_inline243 ) by substituting 0.514809708591 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 0.514809708591 for x, then x=0.514809708591 is a solution.





You can also check your answer by graphing tex2html_wrap_inline255 (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 0.514809708591. This means that 0.514809708591 is the real solution.




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


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Author: Nancy Marcus

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