For a positive number $x$, let the characteristic and mantissa of $\log x$ be $f ( x )$ and $g ( x )$, respectively. The number of natural numbers $n$ satisfying the two inequalities
$$f ( n ) \leq f ( 54 ) , \quad g ( n ) \leq g ( 54 )$$
is? [4 points]
(1) 42
(2) 44
(3) 46
(4) 48
(5) 50

Find the value of $x$ that satisfies the equation $\log _ { 3 } ( x - 11 ) = 3 \log _ { 3 } 2$. [3 points]

What is the value of $\log _ { 2 } 40 - \log _ { 2 } 5$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The temperature of a fire room changes over time. For a certain fire room, let the initial temperature be $T _ { 0 } \left( {}^{\circ}\mathrm{C} \right)$ and the temperature $t$ minutes after the fire starts be $T \left( {}^{\circ}\mathrm{C} \right)$. The following equation holds. $T = T _ { 0 } + k \log ( 8t + 1 ) \quad ($ where $k$ is a constant.$)$ In this fire room with an initial temperature of $20^{\circ}\mathrm{C}$, the temperature was $365^{\circ}\mathrm{C}$ after $\frac{9}{8}$ minutes from the start of the fire, and the temperature was $710^{\circ}\mathrm{C}$ after $a$ minutes from the start of the fire. What is the value of $a$? [3 points]
(1) $\frac{99}{8}$
(2) $\frac{109}{8}$
(3) $\frac{119}{8}$
(4) $\frac{129}{8}$
(5) $\frac{139}{8}$

The temperature of a fire room changes over time. Let the initial temperature of a certain fire room be $T _ { 0 } \left( { } ^ { \circ } \mathrm { C } \right)$, and the temperature $t$ minutes after the fire starts be $T \left( { } ^ { \circ } \mathrm { C } \right)$. The following equation holds. $T = T _ { 0 } + k \log ( 8 t + 1 ) \quad ($ where $k$ is a constant. $)$ In this fire room with an initial temperature of $20 ^ { \circ } \mathrm { C }$, the temperature was $365 ^ { \circ } \mathrm { C }$ at $\frac { 9 } { 8 }$ minutes after the fire started, and the temperature was $710 ^ { \circ } \mathrm { C }$ at $a$ minutes after the fire started. What is the value of $a$? [3 points]
(1) $\frac { 99 } { 8 }$
(2) $\frac { 109 } { 8 }$
(3) $\frac { 119 } { 8 }$
(4) $\frac { 129 } { 8 }$
(5) $\frac { 139 } { 8 }$

For a natural number $n$, $f ( n )$ is defined as follows:
$$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$
For the sequence $\left\{ a _ { n } \right\}$ where $a _ { n } = f \left( 6 ^ { n } \right) - f \left( 3 ^ { n } \right)$, what is the value of $\sum _ { n = 1 } ^ { 15 } a _ { n }$? [3 points]
(1) $120 \left( \log _ { 2 } 3 - 1 \right)$
(2) $105 \log _ { 3 } 2$
(3) $105 \log _ { 2 } 3$
(4) $120 \log _ { 2 } 3$
(5) $120 \left( \log _ { 3 } 2 + 1 \right)$

For a natural number $n$, $f ( n )$ is defined as follows:
$$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$
For two natural numbers $m , n$ with $m, n \leq 20$, how many ordered pairs $( m , n )$ satisfy $f ( m n ) = f ( m ) + f ( n )$?
(1) 220
(2) 230
(3) 240
(4) 250
(5) 260

For a positive real number $x$, let $f ( x )$ and $g ( x )$ be the characteristic and mantissa of $\log x$, respectively. For a natural number $n$, let $a _ { n }$ be the product of all values of $x$ satisfying $f ( x ) - ( n + 1 ) g ( x ) = n$. What is the value of $\lim _ { n \rightarrow \infty } \frac { \log a _ { n } } { n ^ { 2 } }$? [4 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

For a real number $x > 1$, let $f ( x )$ and $g ( x )$ be the characteristic and mantissa of $\log x$ respectively. When the value of $3 f ( x ) + 5 g ( x )$ is a multiple of 10, the values of $x$ are listed in increasing order. Let the 2nd value be $a$ and the 6th value be $b$. What is the value of $\log a b$? [4 points]
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16

In a cylindrical water pipe with cross-sectional radius $R ( R < 1 )$, water flows completely full. Let $v _ { c }$ be the speed of water at the center of the cross-section, and let $v$ be the speed of water at a point $x ( 0 < x \leq R )$ away from the wall toward the center. The following relationship holds: $$\frac { v _ { c } } { v } = 1 - k \log \frac { x } { R }$$ (Here, $k$ is a positive constant, and the unit of length is m and the unit of speed is m/s.) In this water pipe where $R < 1$, when the speed of water at a point $R ^ { \frac { 27 } { 23 } }$ away from the wall toward the center is $\frac { 1 } { 2 }$ of the speed at the center, the speed of water at a point $R ^ { a }$ away from the wall toward the center is $\frac { 1 } { 3 }$ of the speed at the center.
Find the value of $23 a$. [3 points]

What is the value of $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + x ) } { 3 x }$? [2 points]
(1) 1
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 1 } { 5 }$

When compressing digital images, let $P$ denote the peak signal-to-noise ratio, which is an indicator of the difference between the original and compressed images, and let $E$ denote the mean squared error between the original and compressed images. The following relationship holds: $$P = 20 \log 255 - 10 \log E \quad ( E > 0 )$$ When two original images $A$ and $B$ are compressed, let $P _ { A }$ and $P _ { B }$ denote their peak signal-to-noise ratios, and let $E _ { A } \left( E _ { A } > 0 \right)$ and $E _ { B } \left( E _ { B } > 0 \right)$ denote their mean squared errors. If $E _ { B } = 100 E _ { A }$, what is the value of $P _ { A } - P _ { B }$? [3 points]
(1) 30
(2) 25
(3) 20
(4) 15
(5) 10

What is the sum of all natural numbers $x$ that satisfy the exponential inequality $\left( \frac { 1 } { 5 } \right) ^ { 1 - 2 x } \leq 5 ^ { x + 4 }$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

Solve the logarithmic equation $\log _ { 2 } ( x + 6 ) = 5$. [3 points]

When compressing digital images, let $P$ be the peak signal-to-noise ratio, which is an index indicating the degree of difference between the original and compressed images, and let $E$ be the mean squared error between the original and compressed images. The following relationship holds: $$P = 20 \log 255 - 10 \log E \quad ( E > 0 )$$ When two original images $A$ and $B$ are compressed, let the peak signal-to-noise ratios be $P _ { A }$ and $P _ { B }$ respectively, and the mean squared errors be $E _ { A } \left( E _ { A } > 0 \right)$ and $E _ { B } \left( E _ { B } > 0 \right)$ respectively. When $E _ { B } = 100 E _ { A }$, find the value of $P _ { A } - P _ { B }$. [3 points]

For a positive number $x$, let $f ( x )$ be the characteristic (integer part) of $\log x$.
How many natural numbers $n$ not exceeding 100 satisfy $$f ( n + 10 ) = f ( n ) + 1$$ ? [4 points]
(1) 11
(2) 13
(3) 15
(4) 17
(5) 19

For a real number $x \geq \frac { 1 } { 100 }$, let $f ( x )$ be the mantissa of $\log x$. Let $R$ be the region representing the ordered pairs $( a , b )$ of two real numbers satisfying the following conditions on the coordinate plane. (가) $a < 0$ and $b > 10$. (나) The graph of the function $y = 9 f ( x )$ and the line $y = a x + b$ meet at exactly one point.
For a point $( a , b )$ in region $R$, the minimum value of $( a + 20 ) ^ { 2 } + b ^ { 2 }$ is $100 \times \frac { q } { p }$. Find the value of $p + q$. (where $p$ and $q$ are coprime natural numbers.) [4 points]

What is the value of $\log _ { 15 } 3 + \log _ { 15 } 5$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $\lim _ { x \rightarrow 0 } \frac { e ^ { 6 x } - 1 } { \ln ( 1 + 3 x ) }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + 5 x ) } { e ^ { 2 x } - 1 }$? [2 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

For two real numbers $a , b$ greater than 1, $$\log _ { \sqrt { 3 } } a = \log _ { 9 } a b$$ holds. Find the value of $\log _ { a } b$. [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } + 5 x } { \ln ( 1 + 3 x ) }$? [2 points]
(1) $\frac { 7 } { 3 }$
(2) 2
(3) $\frac { 5 } { 3 }$
(4) $\frac { 4 } { 3 }$
(5) 1

For natural numbers $n \geq 2$, what is the sum of all values of $n$ such that $5 \log _ { n } 2$ is a natural number? [4 points]
(1) 34
(2) 38
(3) 42
(4) 46
(5) 50

Let $f ( n )$ denote the number of positive divisors of a natural number $n$, and let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \cdots , a _ { 9 }$ be all positive divisors of 36. What is the value of $\sum _ { k = 1 } ^ { 9 } \left\{ ( - 1 ) ^ { f \left( a _ { k } \right) } \times \log a _ { k } \right\}$? [4 points]
(1) $\log 2 + \log 3$
(2) $2 \log 2 + \log 3$
(3) $\log 2 + 2 \log 3$
(4) $2 \log 2 + 2 \log 3$
(5) $3 \log 2 + 2 \log 3$

For a real number $a$ with $\frac { 1 } { 4 } < a < 1$, let A and B be the points where the line $y = 1$ meets the curves $y = \log _ { a } x$ and $y = \log _ { 4 a } x$ respectively, and let C and D be the points where the line $y = - 1$ meets the curves $y = \log _ { a } x$ and $y = \log _ { 4 a } x$ respectively. Choose all correct statements from the following. [3 points]

$\langle$Statements$\rangle$

ㄱ. The point that divides segment AB externally in the ratio $1 : 4$ has coordinates $( 0,1 )$. ㄴ. If quadrilateral ABCD is a rectangle, then $a = \frac { 1 } { 2 }$. ㄷ. If $\overline { \mathrm { AB } } < \overline { \mathrm { CD } }$, then $\frac { 1 } { 2 } < a < 1$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ