1003138506
Level:
A
Solve.
\[ \frac{\log_3x-3}{3+\log_3x}=0\]
\( x=27 \)
\( x=9 \)
\( x_1=\frac1{27};\ x_2=27 \)
infinitely many solutions
Logarithmic equations and inequalities
1003138505
Level:
A
Solve.
\[ \ln(-2x-6)=\ln(-3x-9) \]
The equation has no solution.
\( x=-3 \)
\( x=3 \)
\( x=-15 \)
1003138504
Level:
A
Solve.
\[ \log_3(-5x-2)=\log_3(x-4) \]
The equation has no solution.
\( x=\frac13 \)
\( x=3 \)
\( x=\frac32 \)
1003138503
Level:
A
How many solutions in the set of integers does the following equation have?
\[ \log_{\frac12}\!(x+6)=\log_{\frac12}\!(3x-6) \]
exactly one positive solution
exactly one negative solution
exactly one solution equal to zero
no solutions
1003138502
Level:
A
Solve.
\[ \log_{\frac13}\!(3-x)=0\]
\( x=2 \)
The equation has no solution.
\( x =3 \)
\( x=-4 \)
1003138501
Level:
A
Solve.
\[ \log_2(3x-5)=4\]
\( x=7 \)
\( x=3 \)
\( x=\frac{11}3 \)
\( x=-\frac13 \)
1003177802
Level:
B
Choose the domain of the expression \( \ln\!\left(-|3-2x|+6\right) \).
\( \left(-\frac32;\frac92\right) \)
\( \left(-\infty;-\frac32\right)\cup\left(\frac92;\infty\right) \)
\( \left(-\infty;\frac32\right)\cup\left(\frac92;\infty\right) \)
\( \left(\frac92;\infty\right) \)
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.
1003143202
Level:
B
What is the total count of all integer solutions of the all following equations?
\[ \begin{aligned}
\log_{\frac12}\!\left(\log_2x\right)&=-1 \\
\log_5\!\left(\log_{\frac15}x\right)&=0 \\
-\log_{\frac13}\!\left(\log_{\frac12}x\right)&=1 \end{aligned}\]
\( 1 \)
\( 2 \)
\( 3 \)
\( 0 \)
1003143201
Level:
B
Solve
\[ \log_3\!\left(3\log_3x+6\right)=1 \text{ .}\]
\( x=\frac13 \)
\( x=27 \)
\( x=9 \)
\( x=\frac19 \)