SOLVING EXPONENTIAL EQUATIONS


Note:

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Solve for the real number x in the following equation.

Problem 7.7 c:

tex2html_wrap_inline728


Answer:
tex2html_wrap_inline155The exact answer is tex2html_wrap_inline730 .

tex2html_wrap_inline155tex2html_wrap_inline155The approximate answers are tex2html_wrap_inline732 and - 6.046449 .


Solution:


Your first objective is to isolate the term tex2html_wrap_inline736 .


Add tex2html_wrap_inline738 to both sides of the equation .


tex2html_wrap_inline728


tex2html_wrap_inline742


tex2html_wrap_inline744


tex2html_wrap_inline746


tex2html_wrap_inline748



Multiply both sides of the equation by tex2html_wrap_inline750 .


tex2html_wrap_inline752


tex2html_wrap_inline754


tex2html_wrap_inline756



Your second objective is to isolate the variable x.


Take the natural logarithm of both sides of the equation tex2html_wrap_inline756 .


tex2html_wrap_inline756


tex2html_wrap_inline762


tex2html_wrap_inline764


tex2html_wrap_inline766


tex2html_wrap_inline768


tex2html_wrap_inline770



Use the Quadratic Formula tex2html_wrap_inline772 where a=1, b=5, tex2html_wrap_inline778 .


tex2html_wrap_inline780


tex2html_wrap_inline782


tex2html_wrap_inline784


tex2html_wrap_inline786


tex2html_wrap_inline788 tex2html_wrap_inline790


tex2html_wrap_inline792 tex2html_wrap_inline794



The exact answers are tex2html_wrap_inline786 and the approximate answers are tex2html_wrap_inline732 and - 6.046449 .



Check this answer in the original equation.



Check the solution tex2html_wrap_inline788 by substituting tex2html_wrap_inline732 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 tex2html_wrap_inline812 for x, then x=1.1092880431 is a solution.


Check the solution tex2html_wrap_inline792 by substituting tex2html_wrap_inline818 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 tex2html_wrap_inline818 for x, then x=-6.046449 is a solution. You can also check your answer by graphing tex2html_wrap_inline828 (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 tex2html_wrap_inline830 This means that tex2html_wrap_inline732 and -6.046449 are the real solutions.








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