a)
log_{\sqrt[3]{3}}9\sqrt{3}=x
(\sqrt[3]{3})^x=9\sqrt{3}
(3^{\frac{1}{3}})^x=3^2\cdot 3^{\frac{1}{2}}
3^{\frac{1}{3}x}=3^{\frac{4}{2}+\frac{1}{2}}
3^{\frac{1}{3}x}=3^{\frac{5}{2}}
\frac{1}{3}x=\frac{5}{2} \ |:\frac{1}{3}
x=\frac{5}{2}\cdot \frac{3}{1}
x=\frac{15}{2}
x=7\frac{1}{2}
log_{\sqrt[3]{3}}9\sqrt{3}=7\frac{1}{2}
b)
log_{\frac{1}{5}}5\sqrt5=x
(\frac{1}{5})^x=5\sqrt{5}
5^{-x}=5^1\cdot 5^{\frac{1}{2}}
5^{-x}=5^{1+\frac{1}{2}}
5^{-x}=5^{1\frac{1}{2}}
-x=1\frac{1}{2} \ |*(-1)
x=-1\frac{1}{2}
log_{\frac{1}{5}}5\sqrt5=-1\frac{1}{2}
c)
log_{\frac{1}{6}}36\sqrt[4]{6}=x
(\frac{1}{6})^x=36\sqrt[4]{6}
6^{-x}=6^2\cdot 6^{\frac{1}{4}}
6^{-x}=6^{2+\frac{1}{4}}
6^{-x}=6^{2\frac{1}{4}}
-x={2\frac{1}{4} \ |*(-1)
x=-2\frac{1}{4}
log_{\frac{1}{6}}36\sqrt[4]{6}=-2\frac{1}{4}
d)
log_{\sqrt3}27\sqrt[4]3=x
(\sqrt{3})^x=27\sqrt[4]{3}
(3^{\frac{1}{2}})^x=3^3\cdot 3^{\frac{1}{4}}
3^{\frac{1}{2}x}=3^{\frac{12}{4}+\frac{1}{4}}
3^{\frac{1}{2}x}=3^{\frac{13}{4}}
\frac{1}{2}x=\frac{13}{4} \ |:\frac{1}{2}
x=\frac{13}{\not4^2}\cdot \frac{\not2^1}{1}=\frac{13}{2}
x=6\frac{1}{2}
log_{\sqrt3}27\sqrt[4]3=6\frac{1}{2}
e)
log_{\frac{1}{2}}16\sqrt[3]{2}=x
(\frac{1}{2})^x=16\sqrt[3]{2}
2^{-x}=2^4\cdot 2^{\frac{1}{3}}
2^{-x}=2^{4+\frac{1}{3}}
2^{-x}=2^{4\frac{1}{3}}
-x=4\frac{1}{3} \ |*(-1)
x=-4\frac{1}{3}
log_{\frac{1}{2}}16\sqrt[3]{2}=-4\frac{1}{3}
f)
log_5125\sqrt{5}=x
5^x=125\sqrt5
5^x=5^3 \cdot 5^{\frac{1}{2}}
5^x=5^{3+\frac{1}{2}}
5^x=5^{3\frac{1}{2}}
x=3\frac{1}{2}
log_5125\sqrt{5}=3\frac{1}{2}
g)
log_{3\sqrt3}81\sqrt[3]{3}=x
(3\sqrt3)^x=81\sqrt[3]{3}
(3^1\cdot 3^{\frac{1}{2}})^x=3^4\cdot 3^{\frac{1}{3}}
(3^{\frac{2}{2}+\frac{1}{2}})^x=3^{\frac{12}{3}+\frac{1}{3}}
3^{\frac{3}{2}x}=3^{\frac{13}{3}}
\frac{3}{2}x=\frac{13}{3} \ |:\frac{3}{2}
x=\frac{13}{3}\cdot \frac{2}{3}
x=\frac{26}{9}
x=2\frac{8}{9}
log_{3\sqrt3}81\sqrt[3]{3}=2\frac{8}{9}