SOLVING LOGARITHMIC EQUATIONS


Note:

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Solve for x in the following equation.


Problem 9.2c:

$8\tan \left( \displaystyle \frac{1}{5}x\right) -14=0$


Answers:
        There are an infinite number of solutions: $x=5\tan
^{-1}\left( \displaystyle \frac{7}{4}\right) \pm n\left( 5\pi \right) $ are the exact solutions, and $x\approx 5.25825\pm n\left( 5\pi \right) $ are the approximate solutions.



Solution:


To solve for x, first isolate the tangent term.

\begin{eqnarray*}&& \\
8\tan \left( \displaystyle \frac{1}{5}x\right) -14 &=&0 ...
...tyle \frac{14}{8}=\displaystyle \frac{7}{4} \\
&& \\
&& \\
&&
\end{eqnarray*}


If we restrict the domain of the cosine function to $-\displaystyle \frac{\pi }{2}<\displaystyle \frac{
1}{5}x<\displaystyle \frac{\...
...}\rightarrow -\displaystyle \frac{5}{2}\leq x\leq \displaystyle \frac{5\pi }{2}$, we can use the arctan function to solve for x.

\begin{eqnarray*}\tan \left( \displaystyle \frac{1}{5}x\right) &=&\displaystyle ...
...ght) &=&\tan
^{-1}\left( \displaystyle \frac{7}{4}\right) \\
&&
\end{eqnarray*}
\begin{eqnarray*}&&\\
\displaystyle \frac{1}{5}x &=&\tan ^{-1}\left( \displayst...
... \frac{7}{4}\right) \\
&& \\
x &\approx &5.25825 \\
&& \\
&&
\end{eqnarray*}


Since the period is $5\pi ,$ this means that the values will repeat every $
5\pi $ radians. Therefore, the solutions are $x=x=5\tan ^{-1}\left( \displaystyle \frac{7
}{4}\right) \pm \pm n\left( 5\pi \right) $ $\approx 5.25825\pm n\left(
15.707963\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=5.25825

Since the left side equals the right side when you substitute 5.25825 for x, then 5.25825 is a solution.




Check the answer $x=5.25825+5\pi =20.966214$

Since the left side equals the right side when you substitute 20.966214for x, then 20.966214 is a solution.




Graphical Check:


Graph the equation

$f(x)=8\tan \left( \displaystyle \frac{1}{5}x\right) -14.$

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


Note the graph crosses at 5.25825 (one of the solutions). Since the period of the function is $5\pi \approx 15.70796$, the graph crosses again at 5.25825+15.70796=2.966214 and again at $
5.25825+2\left( 15.70796\right) \approx 36.674178$, etc.


If you would like to review the solution to problem 9.2d, click on solution.


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