SOLVING LOGARITHMIC EQUATIONS


Note:

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Solve for x in the following equation.


Problem 9.2d:

$\cos \left( \displaystyle \frac{12}{17}x\right) -\displaystyle \frac{2}{17}=0$


Answers:        There are an infinite number of solutions: $x=\displaystyle \frac{17}{12} \cos ^{-1}\left( \displaystyle \frac{2}{17}\right) \pm n\left( \displaystyle \frac{17\pi }{6}\right) $and $x=\displaystyle \frac{17\pi }{6}-\displaystyle \frac{17}{12}\cos ^{-1}\left( \displaystyle \frac{2}{17}\right) \pm n\left( \displaystyle \frac{17\pi }{6}\right) $ are the exact solutions, and $ x\approx 2.05824124\pm n\left( \displaystyle \frac{17\pi }{6}\right) $ and $x\approx 6.84293793\pm n\left( \displaystyle \frac{17\pi }{6}\right) $are the approximate solutions.



Solution:


To solve for x, first isolate the cosine term.

\begin{eqnarray*}&& \\ \cos \left( \displaystyle \frac{12}{17}x\right) -\displa... ...}{17}x\right) &=&\displaystyle \frac{2}{17} \\ && \\ && \\ && \end{eqnarray*}


If we restrict the domain of the cosine function to $0\leq \displaystyle \frac{12}{17} x\leq \pi \rightarrow 0\leq x\leq \displaystyle \frac{17\pi }{12}$, we can use the arccos function to solve for x.

\begin{eqnarray*}\cos \left( \displaystyle \frac{12}{17}x\right) &=&\displaystyl... ...ht) &=&\cos ^{-1}\left( \displaystyle \frac{2}{17}\right) \\ && \end{eqnarray*}
\begin{eqnarray*}&&\\ \displaystyle \frac{12}{17}x &=&\cos ^{-1}\left( \display... ...ac{2}{17}\right) \\ && \\ x &\approx &2.05824124 \\ && \\ && \end{eqnarray*}


The period of $\cos x$ is $2\pi $ and the period of $\cos \left( \displaystyle \frac{12}{ 17}x\right) $ is $\displaystyle \frac{17}{6}\pi .$ The cos x is positive in the firsts and fourth quadrant. This means that the a second solution is $x=\displaystyle \frac{17}{6 }\pi -\cos \left( \displaystyle \frac{12}{17}x\right) .\bigskip\bigskip $

Since the period is $\displaystyle \frac{17}{6}\pi ,$ this means that the values will repeat every $\displaystyle \frac{17}{6}\pi $ radians. Therefore, the solutions are $x= \displaystyle \frac{17}{12}\cos ^{-1}\left( \displaystyle \frac{2}{17}\right) \pm n\left( \displaystyle \frac{17}{6} \pi \right) $ $\approx 2.05824124\pm n\left( 8.901179\right) $, and $x= \displaystyle \frac{17}{6}\pi -\displaystyle \frac{17}{12}\cos ^{-1}\left( \... ...tyle \frac{2}{17}\right) \pm \pm n\left( \displaystyle \frac{17}{6}\pi \right) $ $\approx 6.8429379\pm n\left( 8.901179\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=2.05824124


Since the left side equals the right side when you substitute 2.05824124for x, then 2.05824124 is a solution.




Check the answer x=6.8429379


Since the left side equals the right side when you substitute 6.8429379for x, then 6.8429379 is a solution.



Graphical Check:


Graph the equation

$f(x)=\cos \left( \displaystyle \frac{12}{17}x\right) -\displaystyle \frac{2}{17}.$

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


Note the graph crosses at 2.05824124 (one of the solutions). Since the period of the function is $\displaystyle \frac{17\pi }{6}\approx 8.901179$, the graph crosses again at 2.05824124+8.901179=10.95942 and again at $2.05824124+2\left( 8.901179\right) =19.860599$, etc.

Note the graph crosses at 6.8429379 (one of the solutions). Since the period of the function is $\displaystyle \frac{17\pi }{6}\approx 8.901179$, the graph crosses again at 6.8429379+8.901179=15.7441169 and again at $6.8429379+2\left( 8.901179\right) =24.645296$, etc.

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