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 7.3b:

tex2html_wrap_inline184


Answer:The exact answer is tex2html_wrap_inline186 The approximate answer is tex2html_wrap_inline188 .

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


Solution:


Isolate the exponential term.


Divide both sides of the equation by 23


eqnarray38


eqnarray41





Take the natural logarithm of both sides of the equation tex2html_wrap_inline192


eqnarray55


eqnarray60


eqnarray67


eqnarray67


eqnarray75


eqnarray81


eqnarray87


eqnarray91

The exact answer is tex2html_wrap_inline194 and the approximate answer is tex2html_wrap_inline196





When solving the above problem, you could have used any logarithm. For example, let's solve it using the logarithmic with base 7.


eqnarray55


eqnarray106


eqnarray117


eqnarray125


The exact answer is tex2html_wrap_inline200 and the approximate answer is 0.420217818828.
Although this exact answers looks different from the above exact answer, they are equivalent: both have the same approximate answer.


Check this answer in the original equation.





Check the solution tex2html_wrap_inline194 by substituting 0.420217818828 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 0.420217818828 for x, then x=0.420217818828 is a solution.




You can also check your answer by graphing tex2html_wrap_inline218 (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 0.420217818828. This means that 0.420217818828 is the real solution.




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


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