$$\log _ { 2 } \sqrt { 8 \sqrt { 4 \sqrt { 2 } } }$$
What is the result of this operation?
A) $\frac { 13 } { 8 }$
B) $\frac { 15 } { 8 }$
C) $\frac { 17 } { 8 }$
D) $\frac { 23 } { 16 }$
E) $\frac { 27 } { 16 }$

$x ^ { \ln 4 } - 6 \cdot 2 ^ { \ln x } + 8 = 0$\ What is the product of the $x$ values that satisfy this equation?\ A) $e ^ { 6 }$\ B) $e ^ { 4 }$\ C) $e ^ { 3 }$\ D) $\frac { e ^ { 2 } } { 2 }$\ E) $\frac { e ^ { 3 } } { 3 }$

$\frac { \log _ { 3 } \sqrt { 27 } + \log _ { 27 } \sqrt { 3 } } { \log _ { 3 } \sqrt { 27 } - \log _ { 27 } \sqrt { 3 } }$\ What is the result of this operation?\ A) $\frac { 3 } { 2 }$\ B) $\frac { 4 } { 3 }$\ C) $\frac { 5 } { 4 }$\ D) $\frac { 6 } { 5 }$\ E) $\frac { 7 } { 6 }$

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

On a ruler-like scale with integers from 1 to 50 written on it, the distance of each integer $n$ from 1 is $\log n$ units.
When two identical rulers with this property are placed one below the other as shown in the figure, the number 42 on the upper ruler aligns with the number 28 on the lower ruler, and the number 33 on the upper ruler aligns with the number $x$ on the lower ruler.
Accordingly, what is $x$?
A) 18 B) 19 C) 20 D) 21 E) 22

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) $\frac { 5 } { 4 }$

In a Mathematics lesson, Canan performed operations by following the steps below in order. I. $\operatorname { step } \quad : \quad 6 = 1 \cdot 2 \cdot 3 = \mathrm { e } ^ { \ln 1 } \cdot \mathrm { e } ^ { \ln 2 } \cdot \mathrm { e } ^ { \ln 3 }$ II. $\operatorname { step } : \quad e ^ { \ln 1 } \cdot e ^ { \ln 2 } \cdot e ^ { \ln 3 } = e ^ { \ln 1 + \ln 2 + \ln 3 }$ III. step: $\quad e ^ { \ln 1 + \ln 2 + \ln 3 } = e ^ { \ln 6 }$ IV. $\operatorname { step } : : \mathrm { e } ^ { \ln 6 } = \mathrm { e } ^ { \ln ( 2 + 4 ) }$ V. step: $\mathrm { e } ^ { \ln ( 2 + 4 ) } = \mathrm { e } ^ { \ln 2 + \ln 4 }$ VI. $\operatorname { step } : \quad e ^ { \ln 2 + \ln 4 } = e ^ { \ln 2 } \cdot e ^ { \ln 4 }$ VII. step: $e ^ { \ln 2 } \cdot e ^ { \ln 4 } = 2 \cdot 4 = 8$
At the end of these steps, Canan obtained the result $6 = 8$. Accordingly, in which of the numbered steps did Canan make an error?
A) II
B) III
C) IV
D) V
E) VI

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

Ada calculates the value of $\log _ { 2 } n$ on her scientific calculator for every positive integer n where $\mathrm { n } \leq 32$, and observes that each value is either an integer or a decimal number. Ada writes down either the number itself if the value displayed on the screen is an integer, or the integer part of the number if it is a decimal, and then finds the sum of these numbers she wrote down. Accordingly, what is the result of the sum that Ada found?
A) 94
B) 97
C) 100
D) 103
E) 106

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

On the set of real numbers greater than 1, a function $f$ is defined as
$$f ( x ) = 3 \ln \left( x ^ { 2 } - 1 \right) + 2 \ln \left( x ^ { 3 } - 1 \right) - 5 \ln ( x - 1 )$$
Accordingly,
$$\lim _ { x \rightarrow 1 ^ { + } } e ^ { f ( x ) }$$
what is the value of this limit?
A) 30
B) 36
C) 60
D) 64
E) 72

In the Cartesian coordinate plane, the graphs of functions $f(x) = \log_{2} x$ and $g(x) = \log_{\frac{1}{4}} x$ are given in the figure. Points $A$ and $D$ are on the graph of function $f$, and points $B$ and $C$ are on the graph of function $g$. In the figure, the line segment $[AB]$ passing through the point $(4, 0)$ and the line segment $[CD]$ are both perpendicular to the x-axis, and the area of triangle $ABC$ is 6 square units.
Accordingly, what is the area of triangle $ACD$ in square units?
A) 12 B) 11 C) 10 D) 9 E) 8

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