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 9.1d:         

1.875 sin (x)-0.684=0

Answer:         There are an infinite number of solutions: $x=\sin ^{-1}\left( \displaystyle \frac{228}{625}\right) \pm 2n\pi$ and $x=2\pi -\sin ^{-1}\left( \displaystyle \frac{228 }{625}\right) \pm 2n\pi \$are the exact solutions, and $x\approx 3.3741799\pm 2n\pi$ and x $\approx 2.768175\pm 2n\pi$ are the approximate solutions.

Solution:         To solve for x, first isolate the tangent term.

$$\begin{eqnarray*}&& \\ 1.875sin(x)-0.684 &=&0 \\ && \\ 1.875sin(x) &=&0.684 \... ... \frac{0.684}{1.875}=\displaystyle \frac{228}{625} \\ && \\ && \end{eqnarray*}$$

If we restrict the domain of the sine function to $\left[ -\displaystyle \frac{\pi }{2}, \displaystyle \frac{\pi }{2}\right]$, we can use the arcsine function to solve for x.

$$\begin{eqnarray*}sin(x) &=&\displaystyle \frac{228}{625} \\ && \\ \sin ^{-1}\l... ...228}{625}\right) \\ && \\ x &\approx &0.37341799 \\ && \\ && \end{eqnarray*}$$

The sine function is positive in the first quadrant and the second quadrant. The angle $x=\sin ^{-1}\left( \displaystyle \frac{228}{625}\right)$ is a reference angle that terminates in the first quadrant. The angle that terminates in the second quadrant that has a reference angle $x=\sin ^{-1}\left( \displaystyle \frac{228}{625}\right)$ is $x=\pi -\sin ^{-1}\left( \displaystyle \frac{228}{625}\right) .$

The period of sine function is $2\pi .$ This means that the values will repeat every $2\pi$ radians. Therefore, the exact solutions are $x=\sin ^{-1}\left( \displaystyle \frac{228}{625}\right) \pm 2n\pi$ and $x=\pi -\sin ^{-1}\left( \displaystyle \frac{228}{625}\right) \pm 2n\pi$ where n is an integer. The approximate solutions are $x=0.37341799\pm 2n\pi$ and $x=2.768175\pm 2n\pi$

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=0.37341799

• Left Side: $\qquad 1.875sin(x)-0.684=1.875sin(0.37341799)-0.684\approx 0$

• Right Side:        0

Since the left side equals the right side when you substitute 0.37341799for x, then 0.37341799 is a solution.

Check the answer x=2.768175

• Left Side: $\qquad 1.875sin(x)-0.684=1.875sin(2.768175)-0.684\approx 0$

• Right Side:        0

Since the left side equals the right side when you substitute 2.768175 for x, then 2.768175 is a solution.

Graphical Check:

Graph the equation

f(x)=1.875sin(x)-0.684.

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

Note the graph crosses at 0.37341799 (one of the solutions) and at 2.768175. Since the period of the function is $2\pi \approx 6.283184$, the graph crosses again at $1.5458015\pm 2n\pi$ and again at $2.768175\pm 2n\pi$, etc.

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