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.


Example 4:tex2html_wrap_inline155 tex2html_wrap_inline154


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.


We cannot easily factor this problem. Therefore, let's see if we can use the Quadratic Formula to solve the problem even thought the equation does not look like a quadratic equation. In fact, it is a quadratic equation in tex2html_wrap_inline156


Let's rewrite the equation tex2html_wrap_inline158 with the following substitutions: tex2html_wrap_inline160 and tex2html_wrap_inline162


eqnarray35



Now you should recognize this as a quadratic equation in p where a=1, b=6, and c=-15.


eqnarray42


eqnarray50


eqnarray58



We have two answers for p. However, the original equation did not contain a p. We used p to put the original equation into an equation we could solve. Now, take the p back to tex2html_wrap_inline156


eqnarray66


eqnarray69



Take the natural logarithm of both sides of the equation tex2html_wrap_inline174


eqnarray82


eqnarray88



The exact solution is tex2html_wrap_inline176 and the approximate solution is tex2html_wrap_inline178 This answer may or may not be the solution. You must check it in the original equation.



Check the answers in the original equation.



Check the solution tex2html_wrap_inline176 by substituting 0.641316629016 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 0.641316629016 for x, then x=0.641316629016 is a solution.



You can also check your answer by graphing f(x)= tex2html_wrap_inline194 (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 one place: 0.641316629016. This means that 0.641316629016 is the real solution.








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