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 7.2b:

tex2html_wrap_inline113


Answer: and the approximate answer is tex2html_wrap_inline117



Solution:



The first step is to isolate the exponential term. Therefore, subtract 10 from both sides of the equation tex2html_wrap_inline121

eqnarray28




Take the natural logarithm of both sides of the equation tex2html_wrap_inline123


eqnarray36



eqnarray39



eqnarray42



eqnarray44



The exact answer is tex2html_wrap_inline115 and the approximate answer is tex2html_wrap_inline127




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

eqnarray36



eqnarray51



eqnarray58



eqnarray62



eqnarray68



eqnarray78





Check this answer in the original equation.





Check the solution tex2html_wrap_inline115 by substituting 4.38202663467 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 4.38202663467 for x, then x=4.38202663467 is a solution.





You can also check your answer by graphing tex2html_wrap_inline145 (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 4.38202663467. This means that 4.38202663467 is the real solution.





If you would to review the answer and solution to problem 7.2c, click on Solution.


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Author: Nancy Marcus

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