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 9.1a:         

10 cos $\left( x\right) =8$

Answers:         There are an infinite number of solutions: $x=\cos ^{-1}\left( \displaystyle \frac{4}{5}\right) \pm 2\pi$ and $x=2\pi -\cos ^{-1}\left( \displaystyle \frac{4}{5}\right) \pm 2\pi$ are the exact solutions, and $x\approx 0.6435011\pm 2\pi$ and $x\approx 5.639384\pm 2\pi$ are the approximate solutions.

Solution:

To solve for x, first isolate the cosine term.

$$\begin{eqnarray*}&& \\ 10cos\left( x\right) &=&8 \\ && \\ cos\left( x\right) &=&\displaystyle \frac{8}{10} \\ && \\ && \\ && \end{eqnarray*}$$

If we restrict the domain of the cosine function to $\left[ 0,\pi \right]$, we can use the arccos function to solve for x.

$$\begin{eqnarray*}cos\left( x\right) &=&\displaystyle \frac{8}{10} \\ && \\ \co... ...rac{8}{10}\right) \\ && \\ x &\approx &0.6435011 \\ && \\ && \end{eqnarray*}$$

The cosine of x is positive in the first quadrant and the fourth quadrant. This means that there are two solutions in the first counterclockwise rotation from 0 to $2\pi$. One angle terminates in the first quadrant and the second angle terminates in the fourth quadrant.

We have already determined that the radian measure of the angle that terminates in the first quadrant is $x\approx 0.6435011.$ The radian measure of the angle that terminates in the fourth quadrant is $2\pi -0.6435011=5.639684.$

The period of cosine function is $2\pi .$ This means that the values will repeat every $2\pi$ radians. Therefore, the solutions are $x=0.6435011\pm n\left( 2\pi \right)$ and $x=5.639684\pm n\left( 2\pi \right)$ where n is an integer.

These solutions may or may not be the answers to the original problem. You much check them, either numerically or graphically, with the original equation.

Numerical Check:

Check the answer . x=0.6435011

• Left Side: $\qquad 10cos\left( x\right) =8\approx 10cos\left( 0.6435011\right) \approx 8$

• Right Side:        8

Since the left side equals the right side when you substitute 0.6435011for x, then 0.6435011 is a solution.

Check the answer . x=5.6396841

• Left Side: $\qquad 10cos\left( x\right) =8\approx 10cos\left( 5.639684\right) \approx 8$

• Right Side:        8

Since the left side equals the right side when you substitute 5.6396841for x, then 5.639684 is a solution.

Graphical Check:

Graph the equation

$f(x)=10cos\left( x\right) -8.$

Note that the graph crosses the x-axis many times indicating many solutions.

Note the graph crosses at 0.6435011 ( one of the solutions). Since the period of the function is $2\pi \approx 6.28318530718$, the graph crosses again at 0.6435011+6.28318530718=6.926686 and again at $0.6435011+2\left( 6.28318530718\right) \approx 13.20987$, etc. The graph also crosses at 5.6396841 (another solution we found). Since the period is $2\pi \approx 6.28318530718$, it will crosses again at $5.6396841+2\pi =11.9228694$ and at $5.6396841+2\left( 2\pi \right) \approx 18.20605$, etc.

If you would like to review the solution to problem 9.1b, click on solution.

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