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 9.2c:

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

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

• Left Side: $\qquad 8\tan \left( \displaystyle \frac{1}{5}x\right) -14\approx 8\tan \left( \displaystyle \frac{1}{5}\left( 5.25825\right) \right) -14\approx 0\bigskip$

• Right Side:        $0\bigskip$

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

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

• Left Side: $\qquad 8\tan \left( \displaystyle \frac{1}{5}x\right) -14\approx 8\tan \left( \displaystyle \frac{1}{5}\left( 20.9662145\right) \right) -14\approx 0\bigskip$

• Right Side:        $0\bigskip$

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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