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 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

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) $\log _ { 8 } ( 4 )$

A sequence of five real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 } , a _ { 5 }$ where each term is greater than 1, and between any two adjacent terms, one is twice the other. If $a _ { 1 } = \log _ { 10 } 36$, how many possible values can $a _ { 5 }$ have?
(1) 3
(2) 4
(3) 5
(4) 7
(5) 8

A certain brand of calculator computes the logarithm $\log _ { a } b$ by pressing $\log$（1）$a$（ $b$ ）. A student computing $\log _ { a } b$ (where $a > 1$ and $b > 1$ ) pressed the buttons in the wrong order, pressing $\log$（1）$b$（ $a$ ） instead, obtaining a result that is $\frac { 9 } { 4 }$ times the correct value. Select the relationship between $a$ and $b$.
(1) $a ^ { 2 } = b ^ { 3 }$
(2) $a ^ { 3 } = b ^ { 2 }$
(3) $a ^ { 4 } = b ^ { 9 }$
(4) $2 a = 3 b$
(5) $3 a = 2 b$

Two positive real numbers $a$ and $b$ satisfy $ab^{2} = 10^{5}$ and $a^{2}b = 10^{3}$. Then $\log b = \dfrac{\square}{\square}$. (Express as a fraction in lowest terms)

$$\log _ { 9 } \left( x ^ { 2 } + 2 x + 1 \right) = t \quad ( x > - 1 )$$
Given this equation, which of the following is the expression for x in terms of t?
A) $3 ^ { t } - 1$
B) $3 ^ { \mathrm { t } - 1 }$
C) $3 - 2 ^ { t }$
D) $2 \cdot 3 ^ { \mathrm { t } - 1 }$
E) $3 ^ { t } - 2$

$$\frac{2^{x^{2} - y^{2}}}{4^{x^{2} + xy}} = \frac{1}{2}$$
Given that, what is the value of the expression $(x + y)^{2}$?
A) 2 B) 4 C) 1 D) $\frac{1}{2}$ E) $\frac{1}{4}$

$$\log _ { 2 } 3x + \log _ { 4 } x ^ { 2 } = 2$$
What is the value of x that satisfies the equation?
A) $\frac { \sqrt { 2 } } { 2 }$
B) $\frac { 3 \sqrt { 2 } } { 2 }$
C) $\frac { 5 \sqrt { 2 } } { 2 }$
D) $\frac { \sqrt { 3 } } { 3 }$
E) $\frac { 2 \sqrt { 3 } } { 3 }$

$$\log _ { 2 } \left( \frac { 1 } { \sqrt { x } } \right) + \log _ { 4 } \left( \frac { 4 } { y } \right) = 3$$
Given that, what is the product $x \cdot y$?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 3 } { 8 }$
E) $\frac { 1 } { 16 }$

$\mathbf { x }$ is a real number and
$$\left( \frac { 1 } { 6 } \right) ^ { x } = \left( \frac { 4 } { 3 } \right) ^ { x + 1 }$$
Given this, what is $8 ^ { \mathbf { X } }$?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 1 } { 8 }$
D) $\frac { 3 } { 8 }$
E) $\frac { 2 } { 9 }$

$\log _ { 4 } x$ and $\log _ { 4 } \left( x ^ { 2 } \right)$ are consecutive two positive even integers.
Accordingly, what is the value of $\log _ { x } 4$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 16 }$
D) 1
E) 2

Let k be a positive real number such that for the function
$$f ( x ) = \log _ { x } ( x - k )$$
we have $f ( 3 k ) = \frac { 2 } { 3 }$. What is k?
A) $\frac { 3 } { 8 }$
B) $\frac { 9 } { 8 }$
C) $\frac { 27 } { 8 }$
D) $\frac { 2 } { 9 }$
E) $\frac { 4 } { 9 }$

$\ln x + \ln y = 9$ $$\ln x - \ln y = 3$$ Given this, what is the value of $\log _ { y } x$?\ A) 1\ B) 2\ C) 3\ D) 4\ E) 5

The arithmetic mean of $\log _ { 4 } \mathrm { x }$ and $\log _ { 8 } \frac { 1 } { \mathrm { x } }$ is $\frac { 1 } { 2 }$.
Accordingly, what is the value of $\log _ { 16 } \mathbf { x }$?
A) $\frac { 1 } { 2 }$ B) $\frac { 3 } { 2 }$ C) $\frac { 5 } { 2 }$ D) $\frac { 1 } { 4 }$ E) $\frac { 5 } { 4 }$

Let x be an integer greater than 1.

• $\frac { 64 } { \mathrm { x } }$ is an integer,
• $\frac { \ln 64 } { \ln x }$ is not an integer.

Accordingly, what is the sum of the values that x can take?
A) 40
B) 42
C) 48
D) 54
E) 56

When a stick is divided into 4 equal parts, the length of each part is $\log_5(x)$ units, and when divided into 10 equal parts, the length of each part is $\log_5\left(\frac{x^2}{25}\right)$ units.
Accordingly, what is the length of the stick in units?
A) 5
B) 8
C) 10
D) 12
E) 15

Where $n$ is an integer and $1 < n < 100$,
$$\log_2\left(\log_3 n\right)$$
the value of this expression equals a positive integer. Accordingly, what is the sum of the values that $n$ can take?
A) 36
B) 45
C) 63
D) 72
E) 90

Let $x$ be a positive real number,
$$\log_{4}(x + 5) + \log_{4}(x + 4) - \log_{4}(x + 3) = \log_{2} 3$$
What is the value of $x$ that satisfies this equality?
A) $\sqrt{6}$ B) $\sqrt{7}$ C) $2\sqrt{2}$ D) $2\sqrt{5}$ E) $3\sqrt{2}$

Let $a$, $x$ and $y$ be positive real numbers. The numbers
$$\log_{a} x, \quad \log_{a} y, \quad \log_{a}(x+y)$$
arranged from smallest to largest are consecutive integers. What is the value of $\log_{a}(2a+1)$?
A) $-2$ B) $-1$ C) $2$ D) $3$ E) $4$

Let $a$ and $b$ be non-consecutive positive integers. The equality
$$\ln(a!) = \ln(b!) + 3 \cdot \ln 2 + 2 \cdot \ln 3 + \ln 7$$
is satisfied. Accordingly, what is the sum $a + b$?
A) 10 B) 13 C) 15 D) 18 E) 20

Let $a, b, c$ and $d$ be distinct positive real numbers. The sets $A$ and $B$ are defined as
$$\begin{aligned} & A = \left\{ \log_{2} a, \log_{2} b, \log_{2} c, \log_{2} d \right\} \\ & B = \left\{ \log_{\frac{1}{2}} a, \log_{\frac{1}{2}} b, \log_{\frac{1}{2}} c, \log_{\frac{1}{2}} d \right\} \end{aligned}$$
$$\begin{aligned} & s(A \cap B) = 3 \\ & a \cdot b \cdot c \cdot d = \frac{7}{5} \\ & a + b + c + d = \frac{38}{5} \end{aligned}$$
Given that, what is the sum $a^{2} + b^{2} + c^{2} + d^{2}$?
A) 20 B) 22 C) 24 D) 26 E) 28