${\log }_3{\left(\sqrt{3}\right)}^{15}=15{\log }_3\left(\sqrt{3}\right)\stackrel{\left(\ast \right)}{=}15\cdot \frac{1}{2}=\frac{15}{2}=7,5$  

${\log }_3\left(\sqrt{3}\right)=x$  

$3^x=\sqrt{3}$  

$3^x=3^{\frac{1}{2}}$  

$x=\frac{1}{2}$  

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$\log {\left(\frac{1}{100}\right)}^{\frac{2}{3}}=\frac{2}{3}\log \left(\frac{1}{100}\right)\stackrel{\left(\ast \right)}{=}\frac{2}{3}\cdot \left(-2\right)=-\frac{4}{3}=-1\frac{1}{3}$  

$\log \left(\frac{1}{100}\right)=x$  

${10}^x=\frac{1}{100}$  

${10}^x=\frac{1}{{10}^2}$  

${10}^x={\left(\frac{1}{10}\right)}^2$  

${10}^x={10}^{-2}$  

$x=-2$  

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${\log }_{\frac{1}{2}}{\left(\frac{1}{4}\right)}^{\sqrt{2}}=\sqrt{2}{\log }_{\frac{1}{2}}\left(\frac{1}{4}\right)\stackrel{\left(\ast \right)}{=}\sqrt{2}\cdot 2=2\sqrt{2}$  

${\log }_{\frac{1}{2}}\left(\frac{1}{4}\right)=x$  

${\left(\frac{1}{2}\right)}^x=\left(\frac{1}{4}\right)$  

${\left(\frac{1}{2}\right)}^x=\left(\frac{1}{2^2}\right)$  

${\left(\frac{1}{2}\right)}^x={\left(\frac{1}{2}\right)}^2$  

$x=2$