1. Wykaż, że [latex]-log_{a}b=frac{1}{log_{frac{1}{b}}a}[/latex] 2. liczba [latex]log_{frac{sqrt{2}}{2}}128[/latex] jest równa?

1. [latex]-log_a b=log_a b^{-1} = log_a frac{1}{b} = frac{log_{frac{1}{b}} frac{1}{b}}{log_{frac{1}{b}} a} = frac{1}{log_{frac{1}{b}} a}[/latex]   2. [latex]log_{frac{sqrt{2}}{2}} 128 = x \\ (frac{sqrt{2}}{2})^x = 128 \\ (frac{2^{frac{1}{2}}}{2^1})^x = 2^7 \\ (2^{frac{1}{2}-1})^x = 2^7 \\ (2^{-frac{1}{2}})^x = 2^7 \\ 2^{-frac{1}{2}x} = 2^7 \\ -frac{1}{2}x = 7 / cdot (-2) \\ x = - 14 \\ log_{frac{sqrt{2}}{2}} 128 =-14[/latex]