Calculations with logarithms

1003102414

Level: 
B
From the given expressions choose the one equivalent to \( \log\left( 8\cdot\sqrt[3]{75} \right) \), if \( \log⁡2=a\), \( \log⁡3=b \) and \( \log⁡5=c \).
\( 3a+\frac13 b+\frac23 c \)
\( 3a+\frac13 b+\frac13 c \)
\( 4a+\frac13 b+\frac23 c \)
\( a+\frac13 b+\frac23 c \)

1003102413

Level: 
B
Let \( x \), \( y \), \( z\in (0;\infty) \). Find the equivalent form of the following expression. \[ \log\sqrt{\frac{xz^2}{y^{16}}} \]
\( \frac12\log x-8\log y+\log z \)
\( \frac12\log x+8\log y-\log z \)
\( 8\log x+\frac12\log y-\log z \)
\( \log x-16\log y+2\log z \)

1003102412

Level: 
B
If \( a \), \( b \), \( c\in(0;\infty) \) then the expression \( \log_5⁡a-\frac23 \log_5 b+3\log_5⁡c \) is equivalent to:
\( \log_5\frac{ac^3}{\sqrt[3]{b^2}} \)
\( \log_5⁡\frac{a\sqrt[3]{b^2}}{c^3} \)
\( \log_5⁡\frac{3ac}{\frac23 b} \)
\( \log_5\frac{\frac23 ab}{3c} \)