2000000402
Level:
A
Find the solution of the following equation.
\[
\log_{2x-3}x=1
\]
\(x=3\)
\(x=1\)
\(x=0\)
\(x=-3\)
Logarithmic equations and inequalities
2000000401
Level:
A
Find the solution of the following equation.
\[
\log_{3}(x-9) =2
\]
\(x=18\)
\(x=15\)
\(x=9\)
\(x=0\)
1003138514
Level:
A
How many solutions in the set of integers does the following equation have?
\[ \log_3\frac{2x^2+4}{x^2-4}=1 \]
exactly two solutions
exactly one solution
no solutions
exactly one solution equal to zero
1003138513
Level:
A
How many of the following equations have the solution that is a prime number?
\[ \begin{aligned}
\log_2\!\left(3^x-1\right)&=3 \\
\log_3\!\left(8^x+1\right)&=2 \\
\log_{\frac12}\!\left(2^x+8\right)&=-4 \end{aligned} \]
\( 2 \)
\( 3 \)
\( 1 \)
\( 0 \)
1003138512
Level:
A
How many solutions in the set of integers does the following equation have?
\[ \log_3(x^2-4x)-1=\log_3(2-x) \]
exactly one negative solution
exactly one positive solution
exactly two solutions
no solutions
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.
1003138510
Level:
A
How many solutions does the following equation have?
\[ \log(x-2)+\log(x+2)=2\log(2-x) \]
no solutions
exactly one solution
exactly two solutions
infinitely many solutions
1003138509
Level:
A
Solve.
\[ \log_2(x-1)+\log_2x=1 \]
\( x=2 \)
\( x_1=-1;\ x_2=2 \)
\( x_1=-2;\ x_2=1 \)
\( x=1 \)
1003138508
Level:
A
How many solutions does the following equation have?
\[ 2\log x^4-\log x^5-3\log x^3= 12 \]
exactly one positive solution
exactly one negative solution
exactly one integer solution
no solutions
1003138507
Level:
A
Solve.
\[ \log_4x=2-\log_48\]
\( x=2 \)
\( x=1 \)
\( x=0 \)
The equation has no solution.