Z definicji logarytmu:
[tex]log_{a}b = c \ \ \rightarrow \ \ a^{c} = b[/tex]
[tex]a) \ log_{\frac{1}{2}}\frac{1}{8} = x\\\\(\frac{1}{2})^{x} = \frac{1}{8}\\\\(\frac{1}{2})^{x} = (\frac{1}{2})^{3}\\\\x = 3\\\\\underline{log_{\frac{1}{2}}\frac{1}{8} = 3}[/tex]
[tex]b) \ log_{2}2\sqrt{2} = x\\\\2^{x} = 2\sqrt{2}\\\\2^{x} = 2^{1}\cdot2^{\frac{1}{2}}\\\\2^{x} = 2^{\frac{3}{2}}\\\\x = \frac{3}{2}\\\\\underline{log_{2}2\sqrt{2} = \frac{3}{2}}[/tex]
[tex]c) \ log_{3}9\sqrt{27} = x\\\\3^{x} = 9\sqrt{27}\\\\3^{x} = 3^{2}\cdot\sqrt{3^{3}}\\\\3^{x} = 3^{2}\cdot(3^{3})^{\frac{1}{2}}\\\\3^{x} = 3^{2}\cdot3^{\frac{3}{2}}\\\\3^{x} = 3^{\frac{7}{2}}\\\\x = \frac{7}{2}\\\\\underline{log_{3}9\sqrt{27} = \frac{7}{2}}[/tex]
[tex]d) \ log_{\sqrt{10}}100=x\\\\(\sqrt{10})^{x} = 100\\\\(10^{\frac{1}{2}})^{x} = 10^{2}\\\\10^{\frac{1}{2}x} = 10^{2}\\\\\frac{1}{2}x = 2 \ \ /\cdot2\\\\x = 4\\\\\underline{log_{\sqrt{10}}100 = 4}[/tex]
[tex]e) \ log_{2\sqrt{2}}4 = x\\\\(2\sqrt{2})^{x} = 4\\\\(2\cdot2^{\frac{1}{2}})^{x} = 2^{2}\\\\(2^{\frac{3}{2}})^{x} = 2^{2}\\\\2^{\frac{3}{2}x} = 2^{2}\\\\\frac{3}{2}x = 2 \ \ /\cdot\frac{2}{3}\\\\x = \frac{4}{3}\\\\\underline{log_{2\sqrt{2}}4 = \frac{4}{3}}[/tex]
