oblicz logarytmy

[homerinio]

\(\displaystyle{ a) log _{0,1} 100 = \\
b)log _{2} \frac{1}{ \sqrt{2} } = \\
c)log _{ \frac{1}{2} } 4 = \\
d) log _{9} \frac{1}{3} = \\
e)log _{5} \frac{1}{125} = \\
f) log _{ \frac{2}{3} } \frac{9}{4} \\
g) log _{ \frac{1}{5} } \sqrt[4]{5} = \\
h) log _{1,5} \frac{4}{9}\\
i) log _{ \sqrt{5} } 5=\\
j) log _{ \sqrt{3} } 27=}\)

[rodzyn7773]

\(\displaystyle{ log _{0,1} 100 = log_{ 10^{-1}} 100=c \\ (10^{-1})^{c}=100 \\ 10^{-c}=10^2 \\ -c=2 \\ c=-2}\)

[agulka1987]

\(\displaystyle{ log_{a}b=c \Rightarrow a^c=b}\)

\(\displaystyle{ log_{0,1}100=x}\)

\(\displaystyle{ \left( \frac{1}{10} \right)^x = 100}\)

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

\(\displaystyle{ x=-2}\)


\(\displaystyle{ log_{2} \frac{1}{ \sqrt{2} }=x}\)

\(\displaystyle{ 2^x = \frac{1}{ \sqrt{2} }}\)

\(\displaystyle{ 2^x = 2^{- \frac{1}{2}}}\)

\(\displaystyle{ x=- \frac{1}{2}}\)

pozostałe analogicznie

[rodzyn7773]

\(\displaystyle{ log_{ \frac{2}{3} } \frac{9}{4} =c \\ \left( \frac{2}{3} \right) ^c= \left( \frac{2}{3} \right)^{-2} \\ c=-2}\)