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 7.4b:

tex2html_wrap_inline176


Answer:The exact answer are tex2html_wrap_inline178 and tex2html_wrap_inline180 The approximate answers are tex2html_wrap_inline182 and tex2html_wrap_inline184



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_inline186


eqnarray39


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

eqnarray47


Now we have an equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation tex2html_wrap_inline188


eqnarray59



eqnarray69


Now let's look at the second factor,


eqnarray77


Now we have a second equation where the exponential term is isolated. Take the natural logarithm of both sides of the equation tex2html_wrap_inline190


eqnarray89



eqnarray99



The exact answers are tex2html_wrap_inline178 and tex2html_wrap_inline194 The approximate answers are tex2html_wrap_inline182 and tex2html_wrap_inline184


Check these answers in the original equation.




Check the solution tex2html_wrap_inline178 by substituting 1.2527629685 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 1.2527629685 for x, then x=1.2527629685 is a solution.





Check the solution tex2html_wrap_inline214 by substituting -1.09861228867 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 -1.09861228867 for x, then x=-1.09861228867 is a solution.





You can also check your answer by graphing tex2html_wrap_inline226 (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: -1.09861228867 and 1.2527629685. This means that -1.09861228867 and 1.2527629685 are the real solutions.




If you would like to review the solution to problem 7.4c, click on solution.


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