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 3:

Note that the domain of $\log _{6}\left( 5x\right)$ is the set of real numbers greater than zero because you cannot take the log of zero or a negative number.

The exact value is $x=\displaystyle \displaystyle \frac{6^{8}}{5}=335,923.2.\bigskip\bigskip\bigskip$

Check the solution $x=\displaystyle \displaystyle \frac{6^{8}}{5}$ by substituting x=335,923.2 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: $\qquad \log _{6}\left( 5x\right) =\displaystyle \displaystyle \frac{\ln \left( ... ...displaystyle \frac{\ln \left( 5\cdot 335,923.2\right) }{\ln \left( 6\right) }=8$

• Right Side: $\qquad 8$

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 335,923.2 for x, then x=335,923.2 is a solution.

You can also check your answer by graphing $\quad f(x)=\log _{6}\left( 5x\right) -8\quad$(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 335,923.2. This means that 335,923.2 is the real solution.

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

If you would like to test yourself by working some problems similar to this example, click on Problem.

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, 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