Obliczanie logarytmów

[Mara153]

Oblicz \(\log_{ab}(\sqrt[3]{a} : \sqrt{b})\) jezeli wiadomo, ze \(\log_{ab}a=4\)

[eresh]

Oblicz \(\log_{ab}(\sqrt[3]{a} : \sqrt{b})\) jezeli wiadomo, ze \(\log_{ab}a=4\)

\(\log_{ab}a=4\\
\frac{1}{\log_a(ab)}=4\\
\log_a(ab)=0,25\\
1+\log_ab=0,25\\
\log_ab=-0,75\)

\(\log_{ab}\frac{\sqrt[3]{a}}{\sqrt{b}}=\log_{ab}a^{\frac{1}{3}}-\log_{ab}b^{0,5}=\frac{\frac{1}{3}}{\log_{a}(ab)}-\frac{0,5}{\log_b(ab)}=\frac{\frac{1}{3}}{0,25}-\frac{0,5}{\log_ba+1}=\frac{4}{3}-\frac{0,5}{-\frac{4}{3}+1}=\\=\frac{4}{3}+1,5=\frac{17}{6}\)