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_inline155tex2html_wrap_inline345


The first step is to isolate the expression tex2html_wrap_inline347 .


Subtract 1.4 from both sides of the equation.


tex2html_wrap_inline351


tex2html_wrap_inline353


tex2html_wrap_inline355



Divide both sides of the equation by 2.5.


tex2html_wrap_inline355


tex2html_wrap_inline361


tex2html_wrap_inline363



Take the natural logarithm of both sides of the equation tex2html_wrap_inline363 .


tex2html_wrap_inline363


tex2html_wrap_inline369


tex2html_wrap_inline371


tex2html_wrap_inline373


tex2html_wrap_inline375



Use the Quadratic Formula tex2html_wrap_inline377 where a=1, b=0, tex2html_wrap_inline383 .


tex2html_wrap_inline385


tex2html_wrap_inline387


tex2html_wrap_inline389


tex2html_wrap_inline391 tex2html_wrap_inline393


tex2html_wrap_inline395 tex2html_wrap_inline397



The exact answers are tex2html_wrap_inline389 and the approximate answers are tex2html_wrap_inline401



Check these answers in the original equation.



Check the solution tex2html_wrap_inline391 by substituting tex2html_wrap_inline393 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 2.34963438696 for x, then x=2.34963438696 is a solution.


Check the solution tex2html_wrap_inline395 by substituting tex2html_wrap_inline397 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 -2.34963438696 for x, then x=-2.34963438696 is a solution.


You can also check your answer by graphing tex2html_wrap_inline429 (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 tex2html_wrap_inline431 . This means that 2.34963438696 and -2.34963438696 are the real solutions.








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