1003101103
Level:
C
Find the true statement about the function \( f(x)=\log_2|x| \).
The function \( f \) is even.
The function \( f \) has the minimum at \( x=0 \).
The function \( f \) is bounded.
The function \( f \) is increasing.
Logarithmic functions
1003138403
Level:
C
Find the domain of the function \( f(x)=\sqrt{\log_2x} \).
\( [ 1;\infty ) \)
\( (0;\infty ) \)
\( [ 0;\infty) \)
\( (1;\infty) \)
1003138404
Level:
C
Find the range of the function \( f(x)=\left|\log_3x\right| \).
\( [ 0;\infty ) \)
\( (0;\infty) \)
\( (-\infty;\infty) \)
\( [1;\infty) \)
1003170802
Level:
C
The acidity of a substance is measured by its \( pH \) value. In practice, \( pH \) is often assumed to be the negative logarithm of the hydrogen ion concentration \( H^+ \) (i.e. the number of moles of hydrogen ions per liter of solution):
\[ pH = -\log{(H^+ )}. \]
If an apple juice has a \( pH \) of \( 3.8 \) and a grapefruit juice has a \( pH \) of \( 3.6 \) then the hydrogen ion concentration (\( \mathrm{mol}/\mathrm{l} \)) in the apple juice is how much less than in the grapefruit juice? Round the result to six decimal places.
\( 0.000093 \)
\( -0.000093 \)
\( 1.584893 \)
\( 0 \)
1003170803
Level:
C
Loudness refers to how loud or soft a sound seems to a listener. The loudness \( L \) of sound in decibels, (\( \mathrm{dB} \)) is determined by the formula
\[ L=10\log{\left(\frac I{I_0}\right)}, \]
where \( I \) is intensity of the sound waves (in watts per square meter, \( \mathrm{W}/\mathrm{m}^2 \)) and \( \mathrm{I}_0 \) is fixed and determined intensity of the sound (\( 10^{-12}\,\mathrm{W}/\mathrm{m}^2 \)) that can barely be perceived.
\[ \]
The intensity of the sound produced by the rock band is about \( 100 \) times greater than the intensity of the sound emitted by conventional motor vehicles. How much greater (in \( \mathrm{dB} \)) is loudness of rock band than loudness of motor vehicles?
\( 20 \)
\( 100 \)
\( 10 \)
\( 2 \)
1103082705
Level:
C
Function \( f \) is given completely by the next graph. Identify which of the following statements is false.
\( f(x)=\log_2|x|;\ x\in[0.25;8] \)
\( f(x)=|\log_2 x |;\ x\in[0.25;8] \)
\( f(x)=|-\log_2 x|;\ x\in[0.25;8] \)
\( f(x)=\left|\log_{\frac12} x \right|;\ x\in[0.25;8] \)
1103138401
Level:
C
From the following functions choose the one which is graphed in the picture.
\( f(x)=\left|\log_{\frac12}x\right| \)
\( g(x) = \log_{\frac12}|x| \)
\( k(x)=\log_{\frac12}\sqrt x \)
\( m(x)=\sqrt{\log_{\frac12}x} \)
1103138402
Level:
C
From the following functions choose the one which is graphed in the picture.
\( k(x) = \log_2|x| \)
\( m(x) = |\log_2 x| \)
\( f(x) = \log_2\sqrt x \)
\( g(x) = \sqrt{\log_2 x} \)
1103170801
Level:
C
The acidity of a substance is measured by its \( pH \) value. In practice, \( pH \) is often assumed to be the negative logarithm of the hydrogen ion concentration \( H^+ \) (i.e. the number of moles of hydrogen ions per liter of solution):
\[ pH = -\log{(H^+ )}. \]
In one of the following pictures is a part of the \( pH-H^+ \) graph. Choose the picture.
2010011006
Level:
C
Find the domain of the following function.
\[
f(x)= \log\left( \log_{\frac1{10}}\left(1-x^2\right)\right)
\]
\( (-1 ;0) \cup (0;1)\)
\( (-1 ;1)\)
\( (0;1)\)
\([ 0;1)\)
\((-\infty;1)\)