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 4:

tex2html_wrap_inline132


Note that the domain of tex2html_wrap_inline134 is the set of real numbers such that tex2html_wrap_inline136 or when tex2html_wrap_inline138 because you cannot take the log of zero or a negative number. Note that the expression tex2html_wrap_inline140 is always positive except at tex2html_wrap_inline142


Isolate the logarithmic term.


eqnarray31


eqnarray42


eqnarray52


eqnarray56


eqnarray60



The exact answers are tex2html_wrap_inline146 and the approximate answers are tex2html_wrap_inline148 tex2html_wrap_inline150



Check the answer tex2html_wrap_inline152 by substituting tex2html_wrap_inline154 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 12.5427684616 for x, then tex2html_wrap_inline154 is a solution.


Check the answer tex2html_wrap_inline164 by substituting tex2html_wrap_inline166 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 -7.54276846159 for x, then tex2html_wrap_inline166 is a solution.


You can also check your answer by graphing tex2html_wrap_inline176 (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 12.5427684616 and -7.54276846159. This means that tex2html_wrap_inline154 and -7.54276846159 are the real solutions.



If you have solved the problem by rewriting as , you would have lost one of the solutions.






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