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 9.1b:         

8 sin $\left( x\right) -3=0$

Answer:         There are an infinite number of solutions: $x=\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) \pm 2\pi$ and $x=2\pi -\sin ^{-1}\left( \displaystyle \frac{3}{8} \right) \pm 2\pi$ are the exact solutions, and $x\approx 0.38439677\pm 2\pi$ and $x\approx 2.75719588\pm 2\pi$ are the approximate solutions.

Solution:         To solve for x, you must first isolate the sine term.

$$\begin{eqnarray*}&& \\ 8sin\left( x\right) -3 &=&0 \\ && \\ \sin \left( x\right) &=&\displaystyle \frac{3}{8} \\ && \\ && \\ && \end{eqnarray*}$$

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.

$$\begin{eqnarray*}\sin \left( x\right) &=&\displaystyle \frac{3}{8} \\ && \\ \s... ...rac{3}{8}\right) \\ && \\ x &\approx &0.38439677 \\ && \\ && \end{eqnarray*}$$

The sine of x is positive in the first quadrant and the second quadrant. This means that there are two solutions in the first counterclockwise rotation from 0 to $2\pi$. One angle terminates in the first quadrant and the second angle terminates in the second quadrant.

We have already determined that the radian measure of the angle that terminates in the first quadrant is $x\approx 0.38439677.$ The radian measure of the angle that terminates in the second quadrant is $\pi -0.38439677=2.57519588.$

The period of sine function is $2\pi .$ This means that the values will repeat every $2\pi$ radians. Therefore, the solutions are $x=0.38439677\pm n\left( 2\pi \right)$ and $x=2.75719588\pm n\left( 2\pi \right)$ 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=0.38439677

• Left Side: $\qquad 8sin\left( x\right) -3=8sin\left( 0.38439677\right) -3=-0.0000000333\approx 0$

• Right Side:        0

Since the left side equals the right side when you substitute 0.38439677for x, then 0.38439677 is a solution.

Check the answer x=2.75719588

• Left Side: $\qquad 8sin\left( x\right) -3=8sin\left( 2.75719588\right) -3=-0.0000000672\approx 0$

• Right Side:        0

Since the left side equals the right side when you substitute 2.757195881for x, then 2.75719588 is a solution.

Graphical Check:

Graph the equation

$f(x)=8sin\left( x\right) -3.$

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

Note the graph crosses at 0.38439677 (one of the solutions). Since the period of the function is $2\pi \approx 6.28318530718$, the graph crosses again at 0.38439677+6.28318530718=6.66758 and again at $0.38439677+2\left( 6.28318530718\right) \approx 12.950767$, etc. The graph also crosses at 2.75719588 (another solution we found). Since the period is $2\pi \approx 6.28318530718$, it will cross again at $2.75719588+2\pi =9.0403811$ and at $2.75719588+2\left( 2\pi \right) \approx 15.323566$, etc.

If you would like to review the solution to problem 9.1c, click on solution.

If you would like to go back to the equation table of contents, click on contents.

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