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 $$\frac{17}{6}\pi .$$ 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 $$\frac{17}{6}\pi ,$$ this means that the values will repeat every $$\frac{17}{6}\pi$$ 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

• Left Side: $\qquad \cos \left( \displaystyle \frac{12}{17}x\right) -\displaystyle \frac{2}{... ...\left( 2.05824124\right) \right) -\displaystyle \frac{2}{17} \approx 0\bigskip$

• Right Side:        $0\bigskip$

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

• Left Side: $\qquad \cos \left( \displaystyle \frac{12}{17}x\right) -\displaystyle \frac{2}{... ...}\left( 6.8429379\right) \right) -\displaystyle \frac{2}{17} \approx 0\bigskip$

• Right Side:        $0\bigskip$

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 $$\frac{17\pi }{6}\approx 8.901179$$, 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 $$\frac{17\pi }{6}\approx 8.901179$$, 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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