1003102415
Level:
C
Let \( a \in(0;\infty) \). The expression \( \log_4a-\log_{16}a-\log_{\frac14}a \) is equivalent to:
\( \frac32 \log_4a \)
\( -\frac12\log_4a \)
\( \log_4a \)
\( 0 \)
Calculations with logarithms
1003102414
Level:
B
From the given expressions choose the one equivalent to \( \log\left( 8\cdot\sqrt[3]{75} \right) \), if \( \log2=a\), \( \log3=b \) and \( \log5=c \).
\( 3a+\frac13 b+\frac23 c \)
\( 3a+\frac13 b+\frac13 c \)
\( 4a+\frac13 b+\frac23 c \)
\( a+\frac13 b+\frac23 c \)
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 \( \log_5a-\frac23 \log_5 b+3\log_5c \) is equivalent to:
\( \log_5\frac{ac^3}{\sqrt[3]{b^2}} \)
\( \log_5\frac{a\sqrt[3]{b^2}}{c^3} \)
\( \log_5\frac{3ac}{\frac23 b} \)
\( \log_5\frac{\frac23 ab}{3c} \)
1003102411
Level:
B
Without a calculator evaluate the following expression and select the correct value.
\[ \frac{\log\sqrt6}{\log6} \]
\( \frac12 \)
\( 2 \)
\( \frac{\sqrt6}6 \)
\( \sqrt6-6 \)
1003102410
Level:
C
If \( x\in(0;1)\cup(1;\infty) \) then the product \( \left(\log_x3\right)\left(\log_5x\right) \) can be written as:
\( \log_53 \)
\( \log_35 \)
\( \log_{5x}(3x) \)
\( \log_x3+\log_5x \)
1003102409
Level:
B
Without a calculator evaluate the following expression and select the correct value.
\[ \log_64+\log_69 \]
\( 2 \)
\( 36 \)
\( 13 \)
\( 6 \)
1003102408
Level:
A
The value of the expression \( \log_{16}\left(\log_216 \right) \) is:
\( \frac12 \)
\( \frac14 \)
\( \frac18 \)
\( 2 \)
1003102407
Level:
C
If \( \log_ab=100 \) and \( \log_4a=10 \) then the value of \( b \) is:
\( 2^{2000} \)
\( 2^{1000} \)
\( 2^{1002} \)
\( 2^{200} \)
1003102406
Level:
A
Find the value of \( z \), if
\( \log_3z=-3 \).
\( z=\frac1{27} \)
\( z=27 \)
\( z=-9 \)
\( z=\frac19 \)