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 7.6f: tex2html_wrap_inline155tex2html_wrap_inline275

Answer:

Exact answer: tex2html_wrap_inline155 tex2html_wrap_inline277

Approximate answer: tex2html_wrap_inline155tex2html_wrap_inline279



Solution:


The first step is to isolate tex2html_wrap_inline281


Add tex2html_wrap_inline283 to both sides of the equation.


eqnarray53


eqnarray77



Multiply both sides of the equation by tex2html_wrap_inline285 .


eqnarray102


eqnarray122



The next step is to isolate the variable x.


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


eqnarray136


eqnarray143


eqnarray154


eqnarray164


eqnarray180



The exact answer is tex2html_wrap_inline289 and the approximate answer is tex2html_wrap_inline291



Check the solution tex2html_wrap_inline289 by substituting tex2html_wrap_inline295 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 1.9039 for x, then x=1.9039 is a solution.


You can also check your answer by graphing the function tex2html_wrap_inline305 (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 1.9039. This means that 1.9039 is the real solution.








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