When sound passes through a building wall, a certain proportion is transmitted into the interior while the rest is reflected or absorbed. The ratio of sound transmitted into the interior is called the transmission rate. When the acoustic output of a speaker is $W$ (watts), the intensity $P$ (decibels) of sound transmitted into the interior at a distance of $r$ (m) from the speaker in a building with transmission rate $\alpha$ is as follows. $$\begin{aligned} & P = 10 \log \frac { \alpha W } { I _ { 0 } } - 20 \log r - 11 \\ & \text{(where } I _ { 0 } = 10 ^ { - 12 } \text{ (watts/m}^2\text{) and } r > 1 \text{.)} \end{aligned}$$ A speaker is emitting sound with an acoustic output of 100 (watts). When the intensity of sound transmitted into the interior of a building with transmission rate $\frac { 1 } { 100 }$ is 59 (decibels) or less, what is the minimum distance between the speaker and the building? (Assume that sound spreads uniformly in space and that factors other than transmission rate are not considered.) [4 points]
(1) $10 ^ { 2 } \mathrm{~m}$
(2) $10 ^ { \frac { 17 } { 8 } } \mathrm{~m}$
(3) $10 ^ { \frac { 13 } { 6 } } \mathrm{~m}$
(4) $10 ^ { \frac { 9 } { 4 } } \mathrm{~m}$
(5) $10 ^ { \frac { 5 } { 2 } } \mathrm{~m}$

Solve the system of inequalities $$\left\{ \begin{array} { l } \log _ { 3 } | x - 3 | < 4 \\ \log _ { 2 } x + \log _ { 2 } ( x - 2 ) \geqq 3 \end{array} \right.$$ and find the number of integers $x$ that satisfy it. [3 points]

System of inequalities $$\left\{ \begin{array} { l } \log _ { 3 } | x - 3 | < 4 \\ \log _ { 2 } x + \log _ { 2 } ( x - 2 ) \geqq 3 \end{array} \right.$$ Find the number of integers $x$ that satisfy the system. [3 points]

For a real number $a$ ($a > 1$), let $b = \sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { a } \right) ^ { n }$ be represented as in [Figure 1], and for a real number $c$, let $d = 16 ^ { c }$ be represented as in [Figure 2].
For the real numbers $x$, $y$, $z$ in the figure below, find the value of $\frac { x z } { y }$. [4 points]

Let $a$ be the largest integer among numbers whose common logarithm characteristic is 2, and let $b$ be the smallest number among numbers whose common logarithm characteristic is $-2$. What is the value of $ab$? [4 points]
(1) 0.9
(2) 0.99
(3) 1
(4) 9.99
(5) 10

For a positive number $a$, let the characteristic and mantissa of $\log a$ be $f ( a )$ and $g ( a )$ respectively. Which of the following in are correct? [3 points] 〈Remarks〉 ㄱ. $f ( 2006 ) = 3$ ㄴ. $g ( 2 ) + g ( 6 ) = g ( 12 ) + 1$ ㄷ. If $f ( a b ) = f ( a ) + f ( b )$, then $g ( a b ) = g ( a ) + g ( b )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a positive number $a$, let the characteristic and mantissa of $\log a$ be $f ( a )$ and $g ( a )$ respectively. Which of the following statements in are true? [3 points]

ㄱ. $f ( 2006 ) = 3$ ㄴ. $g ( 2 ) + g ( 6 ) = g ( 12 ) + 1$ ㄷ. If $f ( a b ) = f ( a ) + f ( b )$, then $g ( a b ) = g ( a ) + g ( b )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For two positive numbers $a , b$, $$\left\{ \begin{array} { l } ab = 27 \\ \log _ { 3 } \frac { b } { a } = 5 \end{array} \right.$$ When these conditions hold, find the value of $4 \log _ { 3 } a + 9 \log _ { 3 } b$. [3 points]

For the function with domain $\{ x \mid 1 \leqq x \leqq 81 \}$, $$y = \left( \log _ { 3 } x \right) \left( \log _ { \frac { 1 } { 3 } } x \right) + 2 \log _ { 3 } x + 10$$ Let $M$ be the maximum value and $m$ be the minimum value. Find the value of $M + m$. [4 points]

To remove bacteria living in a water tank, a chemical is to be administered. Let $C _ { 0 }$ be the initial number of bacteria per 1 mL of water in the tank, and let $C$ be the number of bacteria per 1 mL at time $t$ hours after the chemical is administered. The following relationship holds: $$\log \frac { C } { C _ { 0 } } = - k t \quad ( k \text { is a positive constant } )$$ The initial number of bacteria per 1 mL of water is $8 \times 10 ^ { 5 }$, and at time 3 hours after the chemical is administered, the number of bacteria per 1 mL becomes $2 \times 10 ^ { 5 }$. After $a$ hours from administering the chemical, the number of bacteria per 1 mL first becomes $8 \times 10 ^ { 3 }$ or less. Find the value of $a$. (Here, calculate using $\log 2 = 0.3$.) [4 points]

What is the value of $\left( \log _ { 3 } 27 \right) \times 8 ^ { \frac { 1 } { 3 } }$? [2 points]
(1) 12
(2) 10
(3) 8
(4) 6
(5) 4

The value of $\left( \log _ { 3 } 27 \right) \times 8 ^ { \frac { 1 } { 3 } }$ is? [2 points]
(1) 12
(2) 10
(3) 8
(4) 6
(5) 4

For three real numbers $a , b , c$ greater than 1, when $\log _ { a } c : \log _ { b } c = 2 : 1$, what is the value of $\log _ { a } b + \log _ { b } a$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

For an integer $n$, two sets $A ( n ) , B ( n )$ are defined as $$\begin{aligned} & A ( n ) = \left\{ x \mid \log _ { 2 } x \leqq n \right\} \\ & B ( n ) = \left\{ x \mid \log _ { 4 } x \leqq n \right\} \end{aligned}$$ Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $A ( 1 ) = \{ x \mid 0 < x \leqq 1 \}$ ㄴ. $A ( 4 ) = B ( 2 )$ ㄷ. When $A ( n ) \subset B ( n )$, then $B ( - n ) \subset A ( - n )$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ

For the logarithmic equation $\left( \log _ { 2 } x \right) ^ { 2 } - 4 \log _ { 2 } x = 0$, let the two roots be $\alpha , \beta$ respectively. Find the value of $\alpha + \beta$. [3 points]

For $a$ with $0 < a < 1$, when $10 ^ { a }$ is divided by 3, the quotient is an integer and the remainder is 2. What is the sum of all values of $a$? [4 points]
(1) $3 \log 2$
(2) $6 \log 2$
(3) $1 + 3 \log 2$
(4) $1 + 6 \log 2$
(5) $2 + 3 \log 2$

What is the value of $8 ^ { \frac { 2 } { 3 } } + \log _ { 2 } 8$? [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

The value of $8 ^ { \frac { 2 } { 3 } } + \log _ { 2 } 8$ is? [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

The line $y = 2 - x$ intersects the graphs of the two logarithmic functions $y = \log _ { 2 } x$ and $y = \log _ { 3 } x$ at points $\left( x _ { 1 } , y _ { 1 } \right)$ and $\left( x _ { 2 } , y _ { 2 } \right)$, respectively. Which of the following in are correct? [4 points]
ㄱ. $x _ { 1 } > y _ { 2 }$ ㄴ. $x _ { 2 } - x _ { 1 } = y _ { 1 } - y _ { 2 }$ ㄷ. $x _ { 1 } y _ { 1 } > x _ { 2 } y _ { 2 }$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Find the maximum natural number $x$ that satisfies the inequality $\left( \log _ { 3 } x \right) \left( \log _ { 3 } 3 x \right) \leqq 20$. [3 points]

The average number of earthquakes $N$ with magnitude $M$ or greater occurring in a region over one year satisfies the following equation.
$$\log N = a - 0.9 M ( \text{ where } a \text{ is a positive constant } )$$
In this region, earthquakes with magnitude 4 or greater occur on average 64 times per year. Earthquakes with magnitude $x$ or greater occur on average once per year. Find the value of $9 x$. (Use $\log 2 = 0.3$ for the calculation.) [4 points]

For a two-digit natural number $N$, when the mantissa of $\log N$ is $\alpha$,
$$\frac { 1 } { 2 } + \log N = \alpha + \log _ { 4 } \frac { N } { 8 }$$
Find the value of $N$ that satisfies this equation. [4 points]

What is the minimum value of the function $y = 3 + \log _ { 3 } \left( x ^ { 2 } - 4 x + 31 \right)$? [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8

When $a = \log _ { 2 } 10 , b = 2 \sqrt { 2 }$, what is the value of $a \log b$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

For a constant $a$ with $0 < a < \frac { 1 } { 2 }$, let the point where the line $y = x$ meets the curve $y = \log _ { a } x$ be $( p , p )$, and let the point where the line $y = x$ meets the curve $y = \log _ { 2 a } x$ be $( q , q )$. Which of the following statements in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. If $p = \frac { 1 } { 2 }$, then $a = \frac { 1 } { 4 }$. ㄴ. $p < q$ ㄷ. $a ^ { p + q } = \frac { p q } { 2 ^ { q } }$
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ