2010010104
Level:
A
Solve.
\[ \log_3(x-2)+\log_3x=1 \]
\( x=3 \)
\( x_1=3;\ x_2=-1\)
\( x_1=1;\ x_2=-3 \)
\( x=-3 \)
Logarithmic equations and inequalities
1003138511
Level:
B
Solve
\[ \frac{\log(4x)}{\log(x-3)}=\frac{\log16}{\log4} \text{ .} \]
\( x = 9 \)
\( x_1=1;\ x_2=9 \)
\( x_1=-9;\ x_2=-1 \)
The equation has no solution.
1003143201
Level:
B
Solve
\[ \log_3\!\left(3\log_3x+6\right)=1 \text{ .}\]
\( x=\frac13 \)
\( x=27 \)
\( x=9 \)
\( x=\frac19 \)
1003143202
Level:
B
What is the total count of all integer solutions of the all following equations?
\[ \begin{aligned}
\log_{\frac12}\!\left(\log_2x\right)&=-1 \\
\log_5\!\left(\log_{\frac15}x\right)&=0 \\
-\log_{\frac13}\!\left(\log_{\frac12}x\right)&=1 \end{aligned}\]
\( 1 \)
\( 2 \)
\( 3 \)
\( 0 \)
1003143203
Level:
B
Which of the following statements is not true about the equation?
\[ \log_32=\log_3\!\left(3-\log_2x\right) \]
The solution is an odd number.
The solution is \( x=2 \).
The solution is an even number.
The solution is a prime number.
1003158801
Level:
B
Solve
\[ \log_2(x)-\log_4(x)=1\text{ .} \]
\( x=4 \)
\( x=2 \)
\( x_1=\frac12\text{, }x_2=4 \)
\( x_1=-1\text{, }x_2=2 \)
1003158802
Level:
B
Find the solution set of the following equation.
\[ \log_{0.2}(x)-4\log_{0.04}(x)=\log_5(x) \]
\( (0;\infty) \)
\( \mathbb{R} \)
\( \emptyset \)
\( [ 0;\infty) \)
1003158803
Level:
B
Solve
\[ \log_{\frac12}(x)+2=3\log_2(x) \text{ .} \]
\( x=\sqrt2 \)
\( x=2 \)
\( x=-1 \)
Equation has no solution.
1003158804
Level:
B
Which of the following statements is not true about the equation?
\[ \log_x16+\log_{\frac1x}4=2 \]
The solution is an odd number.
The solution is \( x=2 \).
The solution is an even number.
The solution is a prime number.
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.