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 the real number x in the following equation.

Problem 7.6 e:tex2html_wrap_inline155 tex2html_wrap_inline269

Answer:

The exact answer is tex2html_wrap_inline155 tex2html_wrap_inline271

The approximate answer is tex2html_wrap_inline273



Solution:

Your first objective is to isolate the term. tex2html_wrap_inline279

Add tex2html_wrap_inline281 to both sides of the equation .


eqnarray53


eqnarray77



Multiply both sides of the equation by 3.


eqnarray100



Your second objective is to isolate the variable x.


Take the natural logarithm of both sides of the equation tex2html_wrap_inline285


eqnarray121


eqnarray128


eqnarray139


eqnarray149


eqnarray165



The exact answer is tex2html_wrap_inline287 and the approximate answer is tex2html_wrap_inline289



Check this answer in the original equation.



Check the solution tex2html_wrap_inline293 by substituting tex2html_wrap_inline295 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_inline295 for x, then tex2html_wrap_inline305 is a solution.


You can also check your answer by graphing tex2html_wrap_inline307 (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_inline309 This means that tex2html_wrap_inline295 is the real solution.








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