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.

Problem 7.4a:

tex2html_wrap_inline133

Answer:The exact answer are tex2html_wrap_inline135 and tex2html_wrap_inline137 The approximate answers are tex2html_wrap_inline139 and tex2html_wrap_inline141


Solution:

In order to solve this equation, we have to isolate the exponential term. Since we cannot easily do this in the equation's present form, let's tinker with the equation until we have it in a form we can solve.


Factor the left side of the equation tex2html_wrap_inline143

eqnarray36




The only way that a product can equal zero is if at least one of the factors is zero.

eqnarray44




Now we have an equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation tex2html_wrap_inline145

eqnarray51


eqnarray54




Now let's look at the second factor,


eqnarray58




Now we have a second equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation tex2html_wrap_inline147

eqnarray65


eqnarray68



The exact answers are tex2html_wrap_inline135 and tex2html_wrap_inline151 The approximate answers are tex2html_wrap_inline139 and tex2html_wrap_inline141



Check these answers in the original equation.




Check the solution tex2html_wrap_inline135 by substituting 0.693147180567 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 0.69314718056 for x, then x=0.69314718056 is a solution.





Check the solution tex2html_wrap_inline171 by substituting 1.60943791243 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.


  • Left Side: tex2html_wrap_inline175

  • Right Side: tex2html_wrap_inline165


    Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 1.60943791243 for x, then x=1.60943791243 is a solution.





    You can also check your answer by graphing tex2html_wrap_inline183 (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 two places: 0.693147180567 and 1.60943791243. This means that 0.693147180567 and 1.60943791243 are the real solutions.


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