Water flows completely through a cylindrical water pipe with cross-sectional radius $R ( R < 1 )$. 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, the unit of length is m, and the unit of speed is m/s.)
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. What is the value of $a$? [3 points]
(1) $\frac { 39 } { 23 }$
(2) $\frac { 37 } { 23 }$
(3) $\frac { 35 } { 23 }$
(4) $\frac { 33 } { 23 }$
(5) $\frac { 31 } { 23 }$

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

For a natural number $n$, let $a _ { n }$ be the smallest natural number $m$ satisfying the following conditions. What is the value of $\sum _ { n = 1 } ^ { 10 } a _ { n }$? [4 points] (가) The coordinates of point A are $\left( 2 ^ { n } , 0 \right)$. (나) Let D be the point on the line passing through two points $\mathrm { B } ( 1,0 )$ and $\mathrm { C } \left( 2 ^ { m } , m \right)$ whose $x$-coordinate is $2 ^ { n }$. The area of triangle ABD is less than or equal to $\frac { m } { 2 }$.
(1) 109
(2) 111
(3) 113
(4) 115
(5) 117

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 $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + 5 x ) } { e ^ { 2 x } - 1 }$? [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

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

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

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. Which of the following statements in the given options are correct? [4 points]
$\langle$Given Options$\rangle$ ㄱ. The coordinates of the point that divides segment AB externally in the ratio $1 : 4$ are $( 0,1 )$. 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

Find the value of $\log _ { 3 } 72 - \log _ { 3 } 8$. [3 points]

Find the number of natural numbers $n$ such that $\log _ { 4 } 2 n ^ { 2 } - \frac { 1 } { 2 } \log _ { 2 } \sqrt { n }$ is a natural number not exceeding 40. [4 points]

For two constants $a , b$ with $1 < a < b$, the $y$-intercept of the line passing through the two points $\left( a , \log _ { 2 } a \right) , \left( b , \log _ { 2 } b \right)$ and the $y$-intercept of the line passing through the two points $\left( a , \log _ { 4 } a \right) , \left( b , \log _ { 4 } b \right)$ are equal.
For the function $f ( x ) = a ^ { b x } + b ^ { a x }$ with $f ( 1 ) = 40$, what is the value of $f ( 2 )$? [4 points]
(1) 760
(2) 800
(3) 840
(4) 880
(5) 920

Find the value of $\log _ { 2 } 120 - \frac { 1 } { \log _ { 15 } 2 }$. [3 points]

Solve the equation $$\log _ { 2 } ( 3 x + 2 ) = 2 + \log _ { 2 } ( x - 2 )$$ for the real number $x$. [3 points]

For two points $\mathrm{P}(\log_5 3)$ and $\mathrm{Q}(\log_5 12)$ on a number line, the point that divides the line segment PQ internally in the ratio $m:(1-m)$ has coordinate 1. Find the value of $4^m$. (Here, $m$ is a constant with $0 < m < 1$.) [4 points]
(1) $\frac{7}{6}$
(2) $\frac{4}{3}$
(3) $\frac{3}{2}$
(4) $\frac{5}{3}$
(5) $\frac{11}{6}$

For two real numbers $a = 2\log\frac{1}{\sqrt{10}} + \log_{2}20$ and $b = \log 2$, what is the value of $a \times b$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Solve the equation $$\log_{2}(x - 3) = \log_{4}(3x - 5)$$ for the real number $x$. [3 points]