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 7.4g: tex2html_wrap_inline155 tex2html_wrap_inline151



Answer: tex2html_wrap_inline155 No Solution. There is no real number such that tex2html_wrap_inline153

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.


The left side of the equation tex2html_wrap_inline151 is not easily factored. Let's see if we can use the Quadratic Formula.


Note that the equation tex2html_wrap_inline151 can be rewritten as tex2html_wrap_inline159 This is a quadratic equation in tex2html_wrap_inline161 If it is easier for you, substitute a number, say p, in place of tex2html_wrap_inline165 and rewrite the equation tex2html_wrap_inline167 as tex2html_wrap_inline169 let's solve this equation for p.


eqnarray51


eqnarray54


eqnarray62


eqnarray68


However, the initial equation did not contain p, therefore you have to re-substitute tex2html_wrap_inline165 for p and solve for x.



eqnarray77



eqnarray81



eqnarray89



eqnarray96




There are no real answers.


You can also check your answer by graphing tex2html_wrap_inline179 (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 never crosses the x-axis. This means that there are no real solutions.


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