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) $2 + 3 \log 2$

The graph of the logarithmic function $y = \log _ { 2 } ( x + a ) + b$ passes through the focus of the parabola $y ^ { 2 } = x$, and the asymptote of the graph of this logarithmic function coincides with the directrix of the parabola $y ^ { 2 } = x$. What is the value of the sum $a + b$ of the two constants $a , b$? [3 points]
(1) $\frac { 5 } { 4 }$
(2) $\frac { 13 } { 8 }$
(3) $\frac { 9 } { 4 }$
(4) $\frac { 21 } { 8 }$
(5) $\frac { 11 } { 4 }$

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

The line $y = 2 - x$ intersects the graphs of the two logarithmic functions $y = \log _ { 2 } x , y = \log _ { 3 } x$ at points $\left( x _ { 1 } , y _ { 1 } \right), \left( x _ { 2 } , y _ { 2 } \right)$ 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]

In a certain region, the average number of earthquakes $N$ with magnitude $M$ or greater occurring in 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]

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

Let the set $U$ be
$$U = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, a , b , c , d \text { are positive numbers other than } 1 \right\}$$
Let the subset $S$ of $U$ be
$$S = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, \log _ { a } d = \log _ { b } c , \quad a \neq b , \quad b c \neq 1 \right\}$$
Which of the following statements in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. If $A = \left( \begin{array} { l l } 4 & 9 \\ 3 & 2 \end{array} \right)$, then $A \in S$. ㄴ. If $A \in U$ and $A$ has an inverse matrix, then $A \in S$. ㄷ. If $A \in S$, then $A$ has an inverse matrix.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For two real numbers $a , b$ with $1 < a < b$,
$$\frac { 3 a } { \log _ { a } b } = \frac { b } { 2 \log _ { b } a } = \frac { 3 a + b } { 3 }$$
holds. Find the value of $10 \log _ { a } b$. [3 points]

For a natural number $n$, let $f ( n )$ be the mantissa of $\log n$. What is the number of elements in the set
$$A = \{ f ( n ) \mid 1 \leqq n \leqq 150 , n \text { is a natural number } \}$$
? [3 points]
(1) 131
(2) 133
(3) 135
(4) 137
(5) 139

What is the value of $27 ^ { \frac { 1 } { 3 } } + \log _ { 2 } 4$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a natural number $n$ less than 10, when $\left( \frac { n } { 10 } \right) ^ { 10 }$ has a non-zero digit appearing for the first time in the sixth decimal place, what is the value of $n$? (Use $\log 2 = 0.3010 , \log 3 = 0.4771$ for calculations.) [4 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

Find the number of natural numbers $x$ that satisfy the logarithmic inequality $$\log _ { 2 } x \leqq \log _ { 4 } ( 12 x + 28 )$$ [3 points]

What is the value of $4 ^ { \frac { 3 } { 2 } } \times \log _ { 3 } \sqrt { 3 }$? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

What is the value of $4 ^ { \frac { 3 } { 2 } } \times \log _ { 3 } \sqrt { 3 }$? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

To determine the relative density of soil, a test device is inserted into the soil for investigation. When the effective vertical stress of the soil is $S$ and the resistance force received by the test device as it enters the soil is $R$, the relative density $D ( \% )$ of the soil can be calculated as follows. $$D = - 98 + 66 \log \frac { R } { \sqrt { S } }$$ (Here, the units of $S$ and $R$ are metric ton/$\mathrm{m}^{2}$.) The effective vertical stress of soil A is 1.44 times the effective vertical stress of soil B, and the resistance force received by the test device as it enters soil A is 1.5 times the resistance force received as it enters soil B. When the relative density of soil B is $65 ( \% )$, what is the relative density of soil A (in $\%$)? (Use $\log 2 = 0.3$ for calculation.) [3 points]
(1) 81.5
(2) 78.2
(3) 74.9
(4) 71.6
(5) 68.3

To determine the relative density of soil, a method of inserting a test device into the soil for investigation is used. When the effective vertical stress of the soil is $S$ and the resistance force received by the test device as it enters the soil is $R$, the relative density $D ( \% )$ of the soil can be calculated as follows. $$D = - 98 + 66 \log \frac { R } { \sqrt { S } }$$ (where the units of $S$ and $R$ are metric ton $/ \mathrm { m } ^ { 2 }$.) The effective vertical stress of soil A is 1.44 times the effective vertical stress of soil B, and the resistance force received by the test device as it enters soil A is 1.5 times the resistance force received as it enters soil B. When the relative density of soil B is $65 ( \% )$, what is the relative density of soil A (in $\%$)? (Use $\log 2 = 0.3$ for calculation.) [3 points]
(1) 81.5
(2) 78.2
(3) 74.9
(4) 71.6
(5) 68.3

When $\alpha$ is the root of the logarithmic equation $\log _ { 3 } ( x - 4 ) = \log _ { 9 } ( 5 x + 4 )$, find the value of $\alpha$. [3 points]

For a natural number $A$, let the characteristic of $\log A$ be $n$ and the mantissa be $\alpha$. Find the number of values of $A$ such that $n \leqq 2\alpha$ holds. (Given: $3.1 < \sqrt { 10 } < 3.2$) [4 points]

The female silkworm moth secretes pheromone to attract males. When $t$ seconds have passed since the female silkworm moth secreted pheromone, the concentration $y$ of pheromone measured at a distance $x$ from the secretion site satisfies the following equation.
$$\log y = A - \frac { 1 } { 2 } \log t - \frac { K x ^ { 2 } } { t } \text { (where } A \text { and } K \text { are positive constants.) }$$
When 1 second has passed since the female silkworm moth secreted pheromone, the pheromone concentration measured at a distance of 2 from the secretion site is $a$, and when 4 seconds have passed, the pheromone concentration measured at a distance of $d$ from the secretion site is $\frac { a } { 2 }$. What is the value of $d$? [3 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3

The female silkworm moth secretes pheromone to attract males.
When $t$ seconds have passed since the female silkworm moth secreted pheromone, the concentration $y$ of pheromone measured at a distance $x$ from the secretion site satisfies the following equation.
$$\log y = A - \frac { 1 } { 2 } \log t - \frac { K x ^ { 2 } } { t } \text { (where } A \text { and } K \text { are positive constants.) }$$
When 1 second has passed since the female silkworm moth secreted pheromone, the pheromone concentration measured at a distance 2 from the secretion site is $a$, and when 4 seconds have passed, the pheromone concentration measured at a distance $d$ from the secretion site is $\frac { a } { 2 }$. What is the value of $d$? [3 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3