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.


Problem 3:

tex2html_wrap_inline87


Answer: The exact answer is tex2html_wrap_inline89


Solution:


The above equation is valid only if all of the terms are valid. The first term is valid if tex2html_wrap_inline91 tex2html_wrap_inline93 the second term is valid if tex2html_wrap_inline95 the third term is valid if tex2html_wrap_inline97 and the fourth term is valid if tex2html_wrap_inline99 Therefore, the equation is valid when all four of these conditions are met, or when x > 1.5. The domain is the set of real numbers greater than 1.5.




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


eqnarray32


eqnarray34


eqnarray36




Recall that if tex2html_wrap_inline105 then a = b. Therefore, if



eqnarray39



eqnarray41



eqnarray45



The exact answer is tex2html_wrap_inline109





Check the answer x = 14.5 by substituting 14.5 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 14.5 for x, then x = 14.5 is a solution.




You can also check your answer by graphing tex2html_wrap_inline125 (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 14.5. This means that 14.5 is the real solution.



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