2000011302
Level:
B
Find the solution of the equation \(\log_{16}x+\log_4x+\log_2x=7\).
\(x=16\)
\(x=4\)
\(x={\frac12}\)
\(x=2\)
Logarithmic equations and inequalities
2000011301
Level:
B
Let \( x \in (0;1) \cup (1;+\infty)\). Find the value of \(m \in \mathbb{R}\) if \(2\log_m x=\frac32 \log_2 x\).
\(m=2^{\frac43}\)
\(m=2^{\frac34}\)
\(m={\frac34}\)
\(m={\frac43}\)
2010010109
Level:
C
Solve the following inequality.
\[
\log _{0.5}(x+2) < \log _{0.5}8
\]
\(x\in (6;\infty )\)
\(x\in [ 6;\infty )\)
\(x\in (-\infty ;6)\)
\(x\in (0;6 )\)
2010010108
Level:
C
Find the solution set of the following inequality.
\[
\log _{\frac13}(x^{2} - 5x) \geq \log _{\frac13
}6
\]
\([ -1 ;0)\cup (5;6] \)
\((-1 ;0)\cup (5;6)\)
\((-1 ;6)\)
\([ -1 ;6 ] \)
2010010107
Level:
B
Solve the following equation.
\[
\log_2 x^{3}\cdot \log_2 \sqrt[3]{x} +\log_2 \frac{1}
{x} = 6
\]
\(x_{1} = 8\),
\(x_{2} = \frac14\)
\(x_{1} = 2\),
\(x_{2} = 3\)
\(x_{1} = -8\),
\(x_{2} = -\frac14\)
\(x_{1} = \frac18\),
\(x_{2} = 4\)
2010010106
Level:
B
Which of the following statements about the given equation is true?
\[ \log_2(x-2)^2=4-\frac2{\log_2(x-2)} \]
The equation has exactly one solution.
The solution set consists of two prime numbers.
The solution set is the empty set.
None of the above statements is true.
2010010105
Level:
B
Solve.
\[ 3^{2x}=5 \]
\( x=\frac12 \log_3 5 \)
\( x=2 \log_3 5 \)
\( x= \log_3 {5^2} \)
The equation has no solution.
2010010104
Level:
A
Solve.
\[ \log_3(x-2)+\log_3x=1 \]
\( x=3 \)
\( x_1=3;\ x_2=-1\)
\( x_1=1;\ x_2=-3 \)
\( x=-3 \)
2010010103
Level:
B
Solve
\[ \frac{\log(x^2+7)}{\log(x+7)}=\frac{\log{25}}{\log5} \text{ .} \]
\( x=-3 \)
\( x=-5 \)
\( x_1=3;\ x_2=-3 \)
\( x=-2 \)
2010010102
Level:
A
How many solutions in the set of integers does the following equation have?
\[ \log_{2}\!(3x-4)=\log_{2}\!(x-2) \]
no solutions
exactly one solution equal to zero
exactly one negative solution
exactly one positive solution