Solve if log 4 x = -2 find X as a fraction
Solve for x 3^2x-1 = 81^x
find exact values Sin ( 2pi/3) Cos(3pi/4) Tan( 11pi/6)
Please help me!
1. I assume that you mean
$\displaystyle \log_4(x)=-2$ If so: Use the sides of the equation as exponents to the base 4:
$\displaystyle 4^{\log_4(x)}=4^{-2}~\implies~x=4^{-2}~\implies~\boxed{x=\dfrac1{16}}$
2. $\displaystyle 3^{2x-1} = 81^x$
$\displaystyle 3^{2x-1} = (3^4)^x ~\implies~ 3^{2x-1} = 3^{4x}$
Two powers with the same base are equal if the exponents are equal too:
$\displaystyle 2x-1=4x~\implies~x=-\dfrac12$
3. Use a unit-circle to determine the values of the given terms:
$\displaystyle
\begin{aligned} s&=\sin\left(\dfrac23 \pi\right) \cdot \cos\left(\dfrac34 \pi\right) \cdot \tan\left(\dfrac{11}6 \pi\right) \\
&=\dfrac12 \sqrt{3} \cdot \left(-\dfrac12 \sqrt{2}\right) \cdot \left(-\dfrac13 \sqrt{3}\right) \\&= \dfrac14 \sqrt{2}\end{aligned}$