1003159005
Level:
B
Which of the given numbers is the product of all the solutions of the following exponential equation?
\[ x^{\log_3x} =x \]
\( 3 \)
\( 0 \)
\( 1 \)
\( \frac13 \)
Logarithmic equations and inequalities
1003159004
Level:
B
Which of the given numbers is the sum of all the solutions of the equations \( \text{(1)} \) and \( \text{(2)} \)?
\[ \begin{aligned} 10^{x-1}&=2 &\text{(1)} \\
2^{1-x}&=5^x &\text{(2)} \end{aligned} \]
\( \log40 \)
\( \log22 \)
\( \log12+\log_72 \)
\( \log12-\log_72 \)
1003159003
Level:
B
Solve.
\[ 2^x=3^{2-x}\]
\( x=\log_69 \)
\( x=\log_96 \)
\( x=\log_59 \)
\( x=1 \)
1003159002
Level:
B
Solve.
\[ 3^{x-1}=4 \]
\( x=\log_312 \)
\( x=\log_{12}3 \)
\( x=\log_37 \)
\( x=\log4 \)
1003159001
Level:
B
Solve.
\[ 5^{-x}=7 \]
\( x=-\log_57 \)
\( x=-\log_75 \)
\( x=\log_57 \)
The equation has no solution.
1003170904
Level:
C
Find the solution set of the following inequality.
\[ \frac{\log x-1}{2+\log x}\geq1 \]
\( \left(0;\frac1{100}\right) \)
\( \left(0;\frac1{100}\right] \)
\( \left(-\infty;\frac1{100}\right) \)
\( \left(-\infty;\frac1{100}\right] \)
1003170903
Level:
C
Find the solution set of the following inequality.
\[ \log_2^2(x-2)< \log_2(x-2) \]
\( (3;4) \)
\( (0;1) \)
\( (-\infty;4) \)
\( (-\infty;1) \)
1003170902
Level:
C
Find the solution set of the following inequality.
\[ \log_{0.1}\frac{2-x}{x+1}>0 \]
\( \left(\frac12;2\right) \)
\( (-\infty;-1)\cup\left(\frac12;\infty\right) \)
\( \left(-1;\frac12 \right) \)
\( (-1;2) \)
1003170901
Level:
C
Find the solution set of the following inequality.
\[ \log_3\frac{2x-2}{x+3}\leq 0 \]
\( (1;5] \)
\( (-3;5] \)
\( [-3;5] \)
\( [-3;1] \)
1003158805
Level:
B
Which of the following statements about the equation is not true?
\[ \log_{x^2}4+\log_x2+\log_{\frac1x}1=1 \]
The solution is a prime number.
The solution is an even number.
The solution is a positive number.
The solution is an integer.