1103162806
Level:
C
Choose the diagram which shows the number line with the correct solution of the given inequality depicted in blue.
\[ \log_{\frac13}(2-x) < \log_{\frac13}x+\log_{\frac13}3 \]
Logarithmic equations and inequalities
1103162805
Level:
C
Choose the diagram which shows the number line with the correct solution set of the given inequality depicted in blue.
\[ 2\log_{0.1}(x-1) > \log_{0.1}4 \]
1003162804
Level:
C
Find the solution set of the following inequality.
\[ \log_{16} x^2 < \frac12 \]
\( (-2;0)\cup(0;2) \)
\( (-\infty;-2)\cup(2;\infty) \)
\( (0;2) \)
\( (2;\infty) \)
1003162803
Level:
C
Find the solution set of the following inequality.
\[ \log_4(2x-1)\leq 0 \]
\( \left(\frac12;1\right] \)
\( \left(-\infty;1\right] \)
\( \left\{\frac12\right\} \)
\( \left(-\infty;\frac12\right] \)
1003162802
Level:
C
Find the solution set of the following inequality.
\[ \log_{\frac12}(x) \leq 1 \]
\( \left[\frac12;\infty\right) \)
\( \left(-\infty;\frac12\right] \)
\( \left(0;\frac12\right] \)
\( [1;\infty) \)
1003162801
Level:
C
Find the solution set of the following inequality.
\[ \log_{0.3}(x)\geq \log_{0.3}3 \]
\( (0;3] \)
\( (-\infty;3] \)
\( [3;\infty) \)
\( [0;3] \)
1003162704
Level:
B
Which of the following statements about the given equation is true?
\[ \log_4(x-1)^2=3-\frac1{\log_4(x-1)} \]
The solution set consists of two prime numbers.
The solution set is \( \left\{\frac12;1\right\} \).
The solution set is the empty set.
The equation has exactly one solution.
None of the above statements is true.
1003162703
Level:
B
How many solutions does the following equation have?
\[ \ln x^2=\ln^2 x-3 \]
exactly two positive solutions
exactly two solutions, one positive and one negative
no solutions
exactly one solution
1003162702
Level:
B
Which of the given numbers is the sum of all the solutions of the following equation?
\[ \log_2^2 x-3\log_2x+2=0 \]
\( 6 \)
\( 3 \)
\( -3 \)
\( \frac34 \)
1003162701
Level:
B
Find the solution set of the following equation.
\[ \log_3x+\frac3{\log_3x}=4 \]
\( \{3;27\} \)
\( \{1;3\} \)
\( \{-3;-1\} \)
\( \left\{\frac1{27};\frac13\right\} \)