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 7.4f:tex2html_wrap_inline155 tex2html_wrap_inline176

Answer: tex2html_wrap_inline155 The exact answer is tex2html_wrap_inline178 and the approximate answer is tex2html_wrap_inline180


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_inline176 is not easily factored. Let's see if we can use the Quadratic Formula.

Note that the equation tex2html_wrap_inline176 can be rewritten as tex2html_wrap_inline186 This is a quadratic equation in tex2html_wrap_inline188 If it is easier for you, substitute a number, say p, in place of tex2html_wrap_inline192 and rewrite the equation tex2html_wrap_inline194 as tex2html_wrap_inline196 let's solve this equation for p.


eqnarray51


eqnarray54


eqnarray62



However, the initial equation did not contain p, therefore you have to resubstitute tex2html_wrap_inline192 for p and solve for x.


eqnarray71


eqnarray78


eqnarray83


eqnarray92



There is no real number such that tex2html_wrap_inline206 a negative number.


The exact answer is tex2html_wrap_inline208 and the approximate answer is tex2html_wrap_inline180 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_inline208 by substituting -0.352032295516 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 -0.352032295516 for x, then x=0-0.352032295516 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 one place: -0.352032295516. This means that -0.352032295516 is the real solution.




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