[tex]log_{a} c=b~~\Leftrightarrow~~a^{b} =c~~zal.~~a > 0~~\land~~a\neq 1~~\land~~c > 0\\\\a=2~~\land~~b=2,2=\dfrac{11}{5} ~~\Rightarrow~~log_{2} c=\dfrac{11}{5} ~~\Rightarrow~~c=2^{\frac{11}{5} } \\\\a=2~~\land~~b=2,=\dfrac{12}{5} ~~\Rightarrow~~log_{2} c=\dfrac{12}{5} ~~\Rightarrow~~c=2^{\frac{12}{5} } \\\\2,2 < log_{2} 5 < 2,4\\\\\frac{11}{5} < log_{2} 5 < \frac{12}{5}\\\\ log_{2} 2^{\frac{11}{5} } < log_{2} 5 < log_{2}2^{\frac{12}{5} } \\\\2^{\frac{11}{5} } < 5 < 2^{\frac{12}{5} }\\\\[/tex]
[tex]\sqrt[5]{2^{11} } < 5 < \sqrt[5]{2^{12} } ~~\mid ( )^{5} \\\\2^{11} < 5^{5} < 2^{12}\\\\2~048 < 3~125 < 4~096~~nierownosc~~prawdziwa~~\Rightarrow~~log_{2} 5 \in (2,2~;~2,4)[/tex]
