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) $\frac{11}{6}$

Find the value of $\lim_{x \rightarrow 0} \frac{\ln(1+3x)}{\ln(1+5x)}$. [2 points]
(1) $\frac{1}{5}$
(2) $\frac{2}{5}$
(3) $\frac{3}{5}$
(4) $\frac{4}{5}$
(5) 1

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]

Two real numbers $a , b$ greater than 1 satisfy $$\log _ { a } b = 3 , \quad \log _ { 3 } \frac { b } { a } = \frac { 1 } { 2 }$$ What is the value of $\log _ { 9 } a b$? [3 points]
(1) $\frac { 3 } { 8 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 5 } { 8 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 7 } { 8 }$

8. Let $a$ and $b$ be positive numbers not equal to $1$. Then ``$3 ^ { a } > 3 ^ { b } > 3$'' is ``$\log _ { a } 3 < \log _ { b } 3$'' a
(A) necessary and sufficient condition
(B) sufficient but not necessary condition
(C) necessary but not sufficient condition
(D) neither sufficient nor necessary condition

9. Calculate: $\log _ { 2 } \frac { \sqrt { 2 } } { 2 } =$ $\_\_\_\_$ , $2 ^ { \log _ { 2 } 3 + \log _ { 4 } 3 } =$ $\_\_\_\_$.

12. The value of $\lg 0.01 + \log _ { 2 } 16$ is \_\_\_\_.

12. Given $a > 0 , b > 0 , ab = 8$, then when $a$ equals $\_\_\_\_$, $\log _ { 2 } a \cdot \log _ { 2 } ( 2 b )$ attains its maximum value.

12. If $a = \log _ { 2 } 3$ , then $2 ^ { a } + 2 ^ { - a } =$ $\_\_\_\_$ .

If $a > 1$, then the range of values of $x$ satisfying $\log_a(x^2 - 2) < \log_a x$ is
A. $(\sqrt{2}, +\infty)$
B. $(\sqrt{2}, 2)$
C. $(1, \sqrt{2})$
D. $(1, 2)$

Let $x, y, z$ be positive numbers such that $2 ^ { x } = 3 ^ { y } = 5 ^ { z }$, then
A. $2x < 3y < 5z$
B. $5z < 2x < 3y$
C. $3y < 5z < 2x$
D. $3y < 2x < 5z$

Let $a = \log _ { 0.2 } 0.3, b = \log _ { 2 } 0.3$, then
A. $a + b < ab < 0$
B. $ab < a + b < 0$
C. $a + b < 0 < ab$
D. $ab < 0 < a + b$

Given the function $f ( x ) = \log _ { 2 } \left( x ^ { 2 } + a \right)$. If $f ( 3 ) = 1$, then $a = $ \_\_\_\_

In astronomy, the brightness of a celestial body can be described by magnitude or luminosity. The magnitude and luminosity of two stars satisfy $m _ { 2 } - m _ { 1 } = \frac { 5 } { 2 } \lg \frac { E _ { 1 } } { E _ { 2 } }$, where the luminosity of a star with magnitude $m _ { k }$ is $E _ { k } ( k = 1,2 )$. Given that the magnitude of the Sun is $- 26.7$ and the magnitude of Sirius is $- 1.45$, the ratio of the luminosity of the Sun to that of Sirius is (A) $10 ^ { 10.1 }$ (B) 10.1 (C) $\lg 10.1$ (D) $10 ^ { - 10.1 }$

If $a \log _ { 3 } 4 = 2$ , then $4 ^ { - a } =$
A. $\frac { 1 } { 16 }$
B. $\frac { 1 } { 9 }$
C. $\frac { 1 } { 8 }$
D. $\frac { 1 } { 6 }$

Let $a = \log _ { 3 } 2 , b = \log _ { 5 } 3 , c = \frac { 2 } { 3 }$. Then
A. $a < c < b$
B. $a < b < c$
C. $b < c < a$
D. $c < a < b$

Given $5 ^ { 5 } < 8 ^ { 4 } , 13 ^ { 4 } < 8 ^ { 5 }$ . Let $a = \log _ { 5 } 3 , b = \log _ { 8 } 5 , c = \log _ { 13 } 8$ , then
A. $a < b < c$
B. $b < a < c$
C. $b < c < a$
D. $c < a < b$

4. Adolescent vision is a matter of widespread social concern. Vision can be measured using a vision chart. Vision data is usually recorded using the five-point recording method and the decimal recording method. The data $l$ in the five-point recording method and the data $v$ in the decimal recording method satisfy $l = 5 + \log_v$. A student's vision data in the five-point recording method is 4.4. Then their vision data in the decimal recording method is approximately ($\sqrt[10]{10} \approx 1.259$)
A. 1.5
B. 1.2
C. 0.8
D. 0.6