a)
[tex]\log_28\sqrt2-\log_{\frac{1}{2}}\frac{\sqrt2}{4}=\log_2(2^3*2^{\frac{1}{2}})-\log_{\frac{1}{2}}\frac{2^{\frac{1}{2}}}{2^2}=\log_22^{3\frac{1}{2}}-\log_{\frac{1}{2}}2^{-\frac{3}{2}}=\\=3\frac{1}{2}-\log_{\frac{1}{2}}(\frac{1}{2})^{\frac{3}{2}}=3\frac{1}{2}-\frac{3}{2}=2[/tex]
b)
[tex]4*10^{\log3-\log2}+3*10^{\log3+\log2}=4*10^{\log\frac{3}{2}}+3*10^{\log6}=4*\frac{3}{2}+3*6=6+18=24[/tex]
c)
[tex]\frac{\log2}{\log32}+\frac{\log3}{\log243}-\frac{\log4}{\log1024}=\frac{\log2}{\log2^5}+\frac{\log3}{\log3^5}-\frac{\log4}{\log4^5}=\frac{\log2}{5\log2}+\frac{\log3}{5\log3}-\frac{\log4}{5\log4}=\\=\frac{1}{5}+\frac{1}{5}-\frac{1}{5}=\frac{1}{5}[/tex]
