1003159001
Level:
B
Solve.
\[ 5^{-x}=7 \]
\( x=-\log_57 \)
\( x=-\log_75 \)
\( x=\log_57 \)
The equation has no solution.
Logarithmic equations and inequalities
1003159002
Level:
B
Solve.
\[ 3^{x-1}=4 \]
\( x=\log_312 \)
\( x=\log_{12}3 \)
\( x=\log_37 \)
\( x=\log4 \)
1003159003
Level:
B
Solve.
\[ 2^x=3^{2-x}\]
\( x=\log_69 \)
\( x=\log_96 \)
\( x=\log_59 \)
\( x=1 \)
1003159004
Level:
B
Which of the given numbers is the sum of all the solutions of the equations \( \text{(1)} \) and \( \text{(2)} \)?
\[ \begin{aligned} 10^{x-1}&=2 &\text{(1)} \\
2^{1-x}&=5^x &\text{(2)} \end{aligned} \]
\( \log40 \)
\( \log22 \)
\( \log12+\log_72 \)
\( \log12-\log_72 \)
1003159005
Level:
B
Which of the given numbers is the product of all the solutions of the following exponential equation?
\[ x^{\log_3x} =x \]
\( 3 \)
\( 0 \)
\( 1 \)
\( \frac13 \)
1003162701
Level:
B
Find the solution set of the following equation.
\[ \log_3x+\frac3{\log_3x}=4 \]
\( \{3;27\} \)
\( \{1;3\} \)
\( \{-3;-1\} \)
\( \left\{\frac1{27};\frac13\right\} \)
1003162702
Level:
B
Which of the given numbers is the sum of all the solutions of the following equation?
\[ \log_2^2 x-3\log_2x+2=0 \]
\( 6 \)
\( 3 \)
\( -3 \)
\( \frac34 \)
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
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.
1003177802
Level:
B
Choose the domain of the expression \( \ln\!\left(-|3-2x|+6\right) \).
\( \left(-\frac32;\frac92\right) \)
\( \left(-\infty;-\frac32\right)\cup\left(\frac92;\infty\right) \)
\( \left(-\infty;\frac32\right)\cup\left(\frac92;\infty\right) \)
\( \left(\frac92;\infty\right) \)