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 9.2a:

$8\sin\left( 4x\right) -3=0$



Answers:         There are an infinite number of solutions:

$x=\displaystyle \frac{1}{4}
\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) \pm n\left( \displaystyle \frac{\pi }{2}\right) $and $x=\displaystyle \frac{\pi }{4}-\displaystyle \frac{1}{4}\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) \pm
\pi \left( \displaystyle \frac{\pi }{2}\right) $ are the exact solutions, and $x\approx
0.096099\pm n\left( \displaystyle \frac{\pi }{2}\right) $ and $x\approx 0.689299\pm
n\left( \displaystyle \frac{\pi }{2}\right) $ are the approximate solutions.




Solution:


To solve for x, first isolate the sine term.

\begin{eqnarray*}&& \\
8\sin\left( 4x\right) -3 &=&0 \\
&& \\
\sin\left( 4x\right) &=&\displaystyle \frac{3}{8} \\
&& \\
&& \\
&&
\end{eqnarray*}


If we restrict the domain of the cosine function to $-\displaystyle \frac{\pi }{2}\leq
4x\leq \displaystyle \frac{\pi }{2}\rightarrow -\displaystyle \frac{\pi }{8}\leq x\leq \displaystyle \frac{\pi }{8}$, we can use the arcsin function to solve for x.

\begin{eqnarray*}\sin\left( 4x\right) &=&\displaystyle \frac{3}{8} \\
&& \\
\s...
...ht) &=&\sin ^{-1}\left( \displaystyle \frac{3}{8}
\right) \\
&&
\end{eqnarray*}
\begin{eqnarray*}&&\\
4x &=&\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) \...
...ac{3}{8}\right) \\
&& \\
x &\approx &0.096099193 \\
&& \\
&&
\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 4x terminates in the first quadrant and the second angle terminates in the second quadrant. One solution is $x=\displaystyle \frac{1}{4}\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) $


The period of $\sin x$ is $2\pi $, and the period of $\sin 4x$ is $\displaystyle \frac{
\pi }{2}.$ Therefore,the second solution is $x=\displaystyle \frac{\pi }{4}-\displaystyle \frac{1}{4}
\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) \approx 0.689299.\bigskip\bigskip $

Since the period is $\displaystyle \frac{\pi }{2},$ this means that the values will repeat every $\displaystyle \frac{\pi }{2}$ radians. Therefore, the solutions are $
x\approx 0.096099193\pm n\left( \displaystyle \frac{\pi }{2}\right) $ and $x=\pm n\left(
\displaystyle \frac{\pi }{2}\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=0.096099193


Since the left side equals the right side when you substitute 0.096099193for x, then 0.096099193 is a solution.




Check the answer x=0.689299


Since the left side equals the right side when you substitute 0.689299 for x, then 0.689299 is a solution.




Graphical Check:


Graph the equation

$f(x)=8\sin\left( 4x\right) -3.$

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


Note the graph crosses at 0.096099193 ( one of the solutions ). Since the period of the function is $\displaystyle \frac{\pi }{2}\approx
1.5707963$, the graph crosses again at 0.096099193+1.5707963=1.6668955 and again at $0.096099193+2\left( 1.5707963\right) \approx 3.237691793$, etc.

The graph also crosses at 0.689299 ( another solution we found ). Since the period is $\displaystyle \frac{\pi }{2}\approx
1.5707963$, it will crosses again at 0.689299+1.5707963=2.2600953 and at $0.689299+2\left(
1.5707963\right) =3.83089$, etc $.\bigskip\bigskip $


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


If you would like to go back to the previous section, click on previous


If you would like to go to the next section, click on next


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