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:

log $\left( x^{2}-6x-16\right) =5$

The above equation is valid only if

\begin{eqnarray*}&& \\ \left( x^{2}-6x-16\right) &>&0\rightarrow \left( x-8\rig... ...&&or \\ && \\ x-8 &<&0\ \ and\ \ x+2<0\rightarrow x<-2. \\ && \end{eqnarray*}


The domain is the set of real numbers less than -2 or greater than 8.


Convert the equation to an exponential equation with base 10.

\begin{eqnarray*}&& \\ \log \left( x^{2}-6x-16\right) =5 && \\ && \\ && \\ 1... ...& \\ && \\ && \\ 0=x^{2}-6x-\left( 16+10^{5}\right) && \\ && \end{eqnarray*}
\begin{eqnarray*}&& \\ x=\displaystyle \frac{6\pm \sqrt{36+\left( 4\right) \lef... ...& \\ x=\displaystyle \frac{6\pm 2\sqrt{25+10^{5}}}{2} && \\ && \end{eqnarray*}
\begin{eqnarray*}&& \\ x=3\pm \sqrt{100,025} && \\ && \\ && \\ x=3+\sqrt{100... ...\\ x=3-\sqrt{100,025}\approx -313.26729 && \\ && \\ && \\ && \end{eqnarray*}

The exact answers are $\ x=3\pm \sqrt{100,025}$ and the approximate answers are 319.26729 and -313.267290.




These answers may or may not be the solutions. You must check them with the original equation, either by a numerical substitution or by graphing.

Numerical Check:

Check the answer $x=3+\sqrt{100,025}$ by substituting 319.26729 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 319.26729 for x, then x=319.267292 is a solution.



Check the answer $x=3-\sqrt{100025}$ by substituting -313.267290 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.

  • Left Side: $\qquad log\left( x^{2}-6x-16\right) =log\left( \left( -313.267290\right) ^{2}-6\left( -313.267290\right) -16\right) \approx 5 $

  • Right Side:$\qquad 5$

    Since the left side of the original equation is equal to the right side of the original equation after we substitute the value -313.267290 for x, then x=-313.267290 is a solution.




    Graphical Check:

    You can also check your answer by graphing $\quad f(x)=\log \left( x^{2}-6x-16\right) -5\quad $(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 319.267292 and -313.267290. This means that 319.267292 and -313.267290 are the real solutions.

    If you would like to work another example, click on example.


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