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 functions.

Solve for x in the following equation.

Problem 8.5d:

$\log _{_{5}}\left( x+a\right) +\log _{5}\left( x+b\right) =\log _{5}\left( x+c\right)$

Answers: There answers are $x=\displaystyle \frac{-a-b+1\pm \sqrt{ a^{2}-2ab-ca-2b+b^{2}+4x+1}}{2}$.

The restriction on the numbers a, b, and c are

$$\begin{eqnarray*}(1):a^{2}-2ab-ca-2b+b^{2}+4x+1\geq 0 && \\ && \\ (2):\display... ...+1\pm \sqrt{a^{2}-2ab-ca-2b+b^{2}+4x+1}}{2}+c>0 && \\ && \\ && \end{eqnarray*}$$

Simplify the equation and solve.

$$\begin{eqnarray*}&& \\ \log _{_{5}}\left( x+a\right) +\log _{5}\left( x+b\right... ...\left( x+a\right) \left( x+b\right) =\left( x+c\right) && \\ && \end{eqnarray*}$$

$$\begin{eqnarray*}&& \\ x^{2}+ax+bx+ab=x+c && \\ && \\ && \\ x^{2}+ax+bx-x+ab... ... \\ x^{2}+\left( a+b-1\right) x+\left( ab-c\right) =0 && \\ && \end{eqnarray*}$$

$$\begin{eqnarray*}&& \\ x=\displaystyle \frac{-\left( a+b-1\right) \pm \sqrt{\le... ...& \\ && \\ {where }a^{2}-2ab-2a+b^{2}-2b+1+4c &\geq &0, \\ && \end{eqnarray*}$$

$$\begin{eqnarray*}&& \\ \displaystyle \frac{-a-b+1\pm \sqrt{a^{2}-2ab-2a+b^{2}-2... ...sqrt{a^{2}-2ab-2a+b^{2}-2b+1+4c}}{2}+c &>&0 \\ && \\ && \\ && \end{eqnarray*}$$

If you would like to go to the next section, click on next.

If you would like to go back to the previous section, click on previous.

If you would like to go back to the equation table of contents, click on contents.

This site was built to accommodate the needs of students. The topics and problems are what students ask for. We ask students to help in the editing so that future viewers will access a cleaner site. If you feel that some of the material in this section is ambiguous or needs more clarification, or you find a mistake, please let us know by e-mail.

[Algebra] [Trigonometry] [Geometry] [Differential Equations] [Calculus] [Complex Variables] [Matrix Algebra]

S.O.S. MATHematics home page

Copyright © 1999-2004 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA