2022-07-28

1. How many real numbers x satisfy the equation $\frac{1}{5}{\mathrm{log}}_{2}x=\mathrm{sin}\left(5\pi x\right)$?
2. Find the value of x that satisfies ${\mathrm{log}}_{x}25-{\mathrm{log}}_{x}4={\mathrm{log}}_{x}\sqrt{x}$
3. Find the numerical value of ${\mathrm{log}}_{\frac{1}{125}}{25}^{3}\sqrt{25}$.

Hassan Watkins

Expert

How many real numbers x satisfy the equation
${\mathrm{log}}_{2}\left({x}^{1/5}\right)=\mathrm{sin}\left(5\pi x\right)\mathrm{sin}\left(5\pi x\right)$ is always 0.
${2}^{\mathrm{sin}\left(5\pi x\right)}={x}^{1/5}⇒{2}^{0}={x}^{1/5}⇒x=1$
2. Find the value of x that satisfies .
${\mathrm{log}}_{x}\left(25/4\right)={\mathrm{log}}_{x}\left({x}^{1/2}\right)⇒25/4=\sqrt{x}⇒x=625/16$
3. Find the numerical value of.
$={\mathrm{log}}_{{5}^{-3}}{5}^{2}{5}^{2/3}={\mathrm{log}}_{{5}^{-3}}{5}^{8/3}=-8/9$

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