Recall that $x$ is a fixed element of $]0;1[$. Show that:
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \int _ { 0 } ^ { 1 } \left( \frac { t ^ { x - 1 } } { 1 + t } + \frac { t ^ { - x } } { 1 + t } \right) \mathrm { d } t$$

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$ then $x$ equals
(A) $4 ^ { 10 }$
(B) 100
(C) $\log _ { 10 } 4$
(D) none of the above

If $\log_{12} 18 = a$, then $\log_{24} 16$ equals
(A) $\dfrac{8 - 4a}{5 - a}$ (B) $\dfrac{4a - 8}{a - 5}$ (C) $\dfrac{4 + 8a}{5 + a}$ (D) None of these

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$, then $x$ equals
(a) $4 ^ { 10 }$
(b) 100
(c) $\log _ { 10 } 4$
(d) none of the above.

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$, then $x$ equals
(a) $4 ^ { 10 }$
(b) 100
(c) $\log _ { 10 } 4$
(d) none of the above.

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$ then $x$ equals
(A) $4 ^ { 10 }$
(B) 100
(C) $\log _ { 10 } 4$
(D) none of the above

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$ then $x$ equals
(A) $4 ^ { 10 }$
(B) 100
(C) $\log _ { 10 } 4$
(D) none of the above

Let $a, b$ and $c$ be real numbers, each greater than 1, such that $$\frac{2}{3}\log_b a + \frac{3}{5}\log_c b + \frac{5}{2}\log_a c = 3$$ If the value of $b$ is 9, then the value of $a$ must be
(A) $\sqrt[3]{81}$
(B) $\frac{27}{2}$
(C) 18
(D) 27.

Let $p ( n )$ be the number of digits when $8 ^ { n }$ is written in base 6, and let $q ( n )$ be the number of digits when $6 ^ { n }$ is written in base 4. For example, $8 ^ { 2 }$ in base 6 is 144, hence $p ( 2 ) = 3$. Then $\lim _ { n \rightarrow \infty } \frac { p ( n ) q ( n ) } { n ^ { 2 } }$ equals:
(A) 1
(B) $\frac { 4 } { 3 }$
(C) $\frac { 3 } { 2 }$
(D) 2.

The precise interval on which the function $f(x) = \log_{1/2}\left(x^2 - 2x - 3\right)$ is monotonically decreasing, is
(A) $(-\infty, -1)$
(B) $(-\infty, 1)$
(C) $(1, \infty)$
(D) $(3, \infty)$

47. Let $\left( x _ { 0 } , y _ { 0 } \right)$ be the solution of the following equations
$$\begin{aligned} ( 2 x ) ^ { \ln 2 } & = ( 3 y ) ^ { \ln 3 } \\ 3 ^ { \ln x } & = 2 ^ { \ln y } . \end{aligned}$$
Then $x _ { 0 }$ is
(A) $\frac { 1 } { 6 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 1 } { 2 }$
(D) 6

ANSWER: C

1. The value of $\int _ { \sqrt { \ln 2 } } ^ { \sqrt { \ln 3 } } \frac { x \sin x ^ { 2 } } { \sin x ^ { 2 } + \sin \left( \ln 6 - x ^ { 2 } \right) } d x$ is
   (A) $\frac { 1 } { 4 } \ln \frac { 3 } { 2 }$
   (B) $\frac { 1 } { 2 } \ln \frac { 3 } { 2 }$
   (C) $\ln \frac { 3 } { 2 }$
   (D) $\frac { 1 } { 6 } \ln \frac { 3 } { 2 }$

ANSWER: A

The value of $$\left( \left( \log _ { 2 } 9 \right) ^ { 2 } \right) ^ { \frac { 1 } { \log _ { 2 } \left( \log _ { 2 } 9 \right) } } \times ( \sqrt { 7 } ) ^ { \frac { 1 } { \log _ { 4 } 7 } }$$ is $\_\_\_\_$.

Let $m$ be the minimum possible value of $\log _ { 3 } \left( 3 ^ { y _ { 1 } } + 3 ^ { y _ { 2 } } + 3 ^ { y _ { 3 } } \right)$, where $y _ { 1 } , y _ { 2 } , y _ { 3 }$ are real numbers for which $y _ { 1 } + y _ { 2 } + y _ { 3 } = 9$. Let $M$ be the maximum possible value of ( $\log _ { 3 } x _ { 1 } + \log _ { 3 } x _ { 2 } + \log _ { 3 } x _ { 3 }$ ), where $x _ { 1 } , x _ { 2 } , x _ { 3 }$ are positive real numbers for which $x _ { 1 } + x _ { 2 } + x _ { 3 } = 9$. Then the value of $\log _ { 2 } \left( m ^ { 3 } \right) + \log _ { 3 } \left( M ^ { 2 } \right)$ is

Let $e$ denote the base of the natural logarithm. The value of the real number $a$ for which the right hand limit
$$\lim _ { x \rightarrow 0 ^ { + } } \frac { ( 1 - x ) ^ { \frac { 1 } { x } } - e ^ { - 1 } } { x ^ { a } }$$
is equal to a nonzero real number, is $\_\_\_\_$

Let $a = 3 \sqrt { 2 }$ and $b = \frac { 1 } { 5 ^ { 1 / 6 } \sqrt { 6 } }$. If $x , y \in \mathbb { R }$ are such that
$$\begin{aligned} & 3 x + 2 y = \log _ { a } ( 18 ) ^ { \frac { 5 } { 4 } } \\ & 2 x - y = \log _ { b } ( \sqrt { 1080 } ) \end{aligned}$$
then $4 x + 5 y$ is equal to $\_\_\_\_$ .

If the sum of the first 20 terms of the series $\log_{(7^{1/2})}x + \log_{(7^{1/3})}x + \log_{(7^{1/4})}x + \ldots$ is 460, then $x$ is equal to:
(1) $7^2$
(2) $7^{1/2}$
(3) $e^2$
(4) $7^{46/21}$

If for $x \in \left( 0 , \frac { \pi } { 2 } \right) , \log _ { 10 } \sin x + \log _ { 10 } \cos x = - 1$ and $\log _ { 10 } ( \sin x + \cos x ) = \frac { 1 } { 2 } \left( \log _ { 10 } n - 1 \right) , n > 0$, then the value of $n$ is equal to :
(1) 20
(2) 12
(3) 9
(4) 16

If sum of the first 21 terms of the series $\log _ { 9^{1/2} } x + \log _ { 9^{1/3} } x + \log _ { 9^{1/4} } x + \ldots$ where $x > 0$ is 504, then $x$ is equal to
(1) 243
(2) 9
(3) 7
(4) 81

The inverse of $y = 5 ^ { \log x }$ is:
(1) $x = 5 ^ { \log y }$
(2) $x = y ^ { \log 5 }$
(3) $y = x ^ { \frac { 1 } { \log 5 } }$
(4) $x = 5 ^ { \frac { 1 } { \log y } }$

The number of solutions of the equation $\log _ { 4 } ( x - 1 ) = \log _ { 2 } ( x - 3 )$ is

The number of solutions of the equation $\log _ { ( x + 1 ) } \left( 2 x ^ { 2 } + 7 x + 5 \right) + \log _ { ( 2 x + 5 ) } ( x + 1 ) ^ { 2 } - 4 = 0 , x > 0$, is

Let the function $f(x) = \begin{cases} \frac{\log_e(1 + 5x) - \log_e(1 + \alpha x)}{x} & \text{if } x \neq 0 \\ 10 & \text{if } x = 0 \end{cases}$ be continuous at $x = 0$. Then $\alpha$ is equal to
(1) 10
(2) $-10$
(3) 5
(4) $-5$

For three positive integers $p , q , r , x ^ { p q ^ { 2 } } = y ^ { q r } = z ^ { p ^ { 2 } r }$ and $r = p q + 1$ such that 3, $3 \log _ { y } x , 3 \log _ { z } y , 7 \log _ { x } z$ are in A.P. with common difference $\frac { 1 } { 2 }$. The $r - p - q$ is equal to
(1) 2
(2) 6
(3) 12
(4) - 6

Let $a, b, c$ be three distinct positive real numbers such that $2a^{\log_e a} = bc^{\log_e b}$ and $b^{\log_e 2} = a^{\log_e c}$. Then $6a + 5bc$ is equal to $\_\_\_\_$.

The sum of all the solutions of the equation $( 8 ) ^ { 2 x } - 16 \cdot ( 8 ) ^ { x } + 48 = 0$ is :
(1) $1 + \log _ { 8 } ( 6 )$
(2) $1 + \log _ { 6 } ( 8 )$
(3) $\log _ { 8 } ( 6 )$
(4) $\log _ { 8 } ( 4 )$