SOLVING LOGARITHMIC EQUATIONS


Note:

If you would like an in-depth review of logarithms, the rules of logarithms, logarithmic functions and logarithmic equations, click on logarithmic function.


Solve for x in the following equation.


Example 2:

tex2html_wrap_inline122

The above equation is valid only if all of the terms are valid. The first term is valid if 3x- 5 > 0 or tex2html_wrap_inline126 the second term is valid if 2x- 3 > 0 or tex2html_wrap_inline130 and the third term is valid if x > 0. Therefore, the equation is valid when all three of these conditions are met, or when . The domain is the set of real numbers greater than tex2html_wrap_inline136.



Simplify both sides of the equation using the rules of logarithms.

eqnarray28




Recall that if tex2html_wrap_inline138 then a=b . Therefore, if


eqnarray32


eqnarray36



Solve for x.


eqnarray42


eqnarray45


eqnarray49


eqnarray53


eqnarray57


eqnarray61



The exact answer is tex2html_wrap_inline142 and the approximate answer is tex2html_wrap_inline144




Check the answer tex2html_wrap_inline146 by substituting 8.63104367407 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 8.63104367407 for x, then tex2html_wrap_inline158 is a solution.




You can also check your answer by graphing tex2html_wrap_inline160 (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 8.63104367407. This means that 8.63104367407 is the real solution.



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