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 functions. under Algebra.

Solve for x in the following equation.


Problem 7.4e: tex2html_wrap_inline177


Answer: tex2html_wrap_inline155 The exact answer is tex2html_wrap_inline179 and the approximate answer is tex2html_wrap_inline181


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_inline177 is not easily factored. Let's see if we can use the Quadratic Formula.

Note that the equation tex2html_wrap_inline177 can be rewritten as tex2html_wrap_inline187 This is a quadratic equation in tex2html_wrap_inline189 If it is easier for you, substitute a number, say p, in place of tex2html_wrap_inline193 and rewrite the equation tex2html_wrap_inline195 as tex2html_wrap_inline197 let's solve this equation for p.


eqnarray50


eqnarray53


eqnarray57


eqnarray63



However, the initial equation did not contain p, therefore you have to resubstitute tex2html_wrap_inline193 for p and solve for x.


eqnarray72


eqnarray79


eqnarray86


eqnarray93


There is no real number such that tex2html_wrap_inline207 a negative number.


The exact answer is tex2html_wrap_inline209 and the approximate answer is tex2html_wrap_inline211 These answers may or may not be solutions to the original equations. You must check the answers in the original equation.



Check this answer in the original equation.


Check the solution tex2html_wrap_inline215 by substituting 0.762831450377 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 0.762831450377 for x, then x=0.762831450377 is a solution.



You can also check your answer by graphing tex2html_wrap_inline227 (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: 0.762831450377. This means that 0.762831450377 is the real solution.




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