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 7.7a:

tex2html_wrap_inline319


Answer:


Solution:


The first step is to isolate tex2html_wrap_inline327 .


Subtract 14 from both sides of the equation.


tex2html_wrap_inline319


tex2html_wrap_inline333


tex2html_wrap_inline335



Divide both sides of the equation by 2.


tex2html_wrap_inline335


tex2html_wrap_inline341


tex2html_wrap_inline343



The next step is to isolate the variable x.


Take the natural logarithm of both sides of the equation..


tex2html_wrap_inline343


tex2html_wrap_inline349


tex2html_wrap_inline351


tex2html_wrap_inline353


tex2html_wrap_inline355



Use the Quadratic Formula tex2html_wrap_inline357 where a=2, b=5, tex2html_wrap_inline363


tex2html_wrap_inline365


tex2html_wrap_inline367


tex2html_wrap_inline369


tex2html_wrap_inline371


tex2html_wrap_inline373


tex2html_wrap_inline375 tex2html_wrap_inline377


tex2html_wrap_inline379 tex2html_wrap_inline381



The exact answers are tex2html_wrap_inline373 and the approximate answers are tex2html_wrap_inline385 and - 3.618639.


Check the solution tex2html_wrap_inline375 by substituting 1.118639 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 1.118639 for x, then x=1.118639 is a solution.


Check the solution tex2html_wrap_inline379 by substituting -3.618639 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 -3.618639 for x, then x=-3.618639 is a solution.


You can also check your answer by graphing (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 1.118639 and -3.618639. This means that 1.118639 and -3.618639 are the real solutions.








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