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. under Algebra.

Solve for x in the following equation.

Example 1:tex2html_wrap_inline155tex2html_wrap_inline189

Isolate the exponential term.

Divide both sides of the equation by 4


eqnarray25



Take the natural logarithm of both sides of the equation tex2html_wrap_inline191


eqnarray40


eqnarray46


eqnarray53


eqnarray55



The exact answer is tex2html_wrap_inline193 ( which can also be written tex2html_wrap_inline195 ) and the approximate answer is tex2html_wrap_inline197



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


eqnarray40


eqnarray76


eqnarray87


eqnarray99


eqnarray108


eqnarray117


eqnarray126


eqnarray138



Check this answer in the original equation.



Check the solution tex2html_wrap_inline203


(can also be written in the equivalent form tex2html_wrap_inline205 )


by substituting 0.111571775657 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 0.111571775657 for x, then x=0.111571775657 is a solution.


You can also check your answer by graphing tex2html_wrap_inline217 (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 0.111571775657. This means that 0.111571775657 is the real solution.








If you would like to work another example, click on Example


If you would like to test yourself by working some problems similar to this example, click on Problem


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Author: Nancy Marcus

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