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

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) $\frac { 7 } { 8 }$

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

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

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

103- What is the value of $\log_{21}(1323) + \log_{21}(147)\log_{21}(3) + \left(\log_{21}(3)\right)^2$?
(1) $1$ (2) $2$ (3) $3$ (4) $4$

Let $p ( n )$ be the number of digits when $8 ^ { n }$ is written in base 6, and let $q ( n )$ be the number of digits when $6 ^ { n }$ is written in base 4. For example, $8 ^ { 2 }$ in base 6 is 144, hence $p ( 2 ) = 3$. Then $\lim _ { n \rightarrow \infty } \frac { p ( n ) q ( n ) } { n ^ { 2 } }$ equals:
(A) 1
(B) $\frac { 4 } { 3 }$
(C) $\frac { 3 } { 2 }$
(D) 2.

The value of $$\left( \left( \log _ { 2 } 9 \right) ^ { 2 } \right) ^ { \frac { 1 } { \log _ { 2 } \left( \log _ { 2 } 9 \right) } } \times ( \sqrt { 7 } ) ^ { \frac { 1 } { \log _ { 4 } 7 } }$$ is $\_\_\_\_$.

$\lim _ { x \rightarrow 0 } \left( \tan \left( \frac { \pi } { 4 } + x \right) \right) ^ { 1 / x }$ is equal to
(1) $e$
(2) 2
(3) 1
(4) $e ^ { 2 }$

Let $f ( x ) = \lim _ { \mathrm { n } \rightarrow \infty } \sum _ { \mathrm { r } = 0 } ^ { \mathrm { n } } \left( \frac { \tan \left( x / 2 ^ { r + 1 } \right) + \tan ^ { 3 } \left( x / 2 ^ { r + 1 } \right) } { 1 - \tan ^ { 2 } \left( x / 2 ^ { r + 1 } \right) } \right)$. Then $\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { f ( x ) } } { ( x - f ( x ) ) }$ is equal to

If $\lim _ { \mathrm { t } \rightarrow 0 } \left( \int _ { 0 } ^ { 1 } ( 3 x + 5 ) ^ { \mathrm { t } } \mathrm { d } x \right) ^ { \frac { 1 } { t } } = \frac { \alpha } { 5 \mathrm { e } } \left( \frac { 8 } { 5 } \right) ^ { \frac { 2 } { 3 } }$, then $\alpha$ is equal to $\_\_\_\_$

Let $a_1, a_2, a_3, a_4$ be a geometric sequence with first term 10 and common ratio 10. Let $b = \sum_{n=1}^{3} \log_{a_n} a_{n+1}$. Select the correct option.
(1) $2 < b \leq 3$
(2) $3 < b \leq 4$
(3) $4 < b \leq 5$
(4) $5 < b \leq 6$
(5) $6 < b \leq 7$

On a coordinate plane, it is known that vector $\overrightarrow{PQ} = \left(\log \frac{1}{5}, -10^{-5}\right)$, where point $P$ has coordinates $\left(\log \frac{1}{2}, 2^{-5}\right)$. Select the correct option.
(1) Point $Q$ is in the first quadrant
(2) Point $Q$ is in the second quadrant
(3) Point $Q$ is in the third quadrant
(4) Point $Q$ is in the fourth quadrant
(5) Point $Q$ is on a coordinate axis

Select the value of $\sum _ { k = 1 } ^ { 5 } \log _ { 7 } \left( \frac { 2 k - 1 } { 2 k + 1 } \right)$.
(1) $- \log 11$
(2) $\log 11$
(3) $\log \frac { 11 } { 7 }$
(4) $- \frac { \log 11 } { \log 7 }$
(5) $\frac { \log 11 } { \log 7 }$

5. Using the observation that $2 ^ { 5 } \approx 3 ^ { 3 }$, it is possible to deduce that $\log _ { 3 } 2$ is approximately
A $\frac { 3 } { 5 }$
B $\frac { 2 } { 3 }$
C $\quad \frac { 3 } { 2 }$
D $\frac { 5 } { 3 }$
E $\frac { 1 } { 2 }$
F 2

A student wishes to evaluate the function $\mathrm { f } ( x ) = x \sin x$, where $x$ is in radians, but has a calculator that only works in degrees.
What could the student type into their calculator to correctly evaluate $\mathrm { f } ( 4 )$ ?
A $( \pi \times 4 \div 180 ) \times \sin ( 4 )$
B $( \pi \times 4 \div 180 ) \times \sin ( \pi \times 4 \div 180 )$
C $4 \times \sin ( \pi \times 4 \div 180 )$
D $( 180 \times 4 \div \pi ) \times \sin ( 4 )$
E $\quad ( 180 \times 4 \div \pi ) \times \sin ( 180 \times 4 \div \pi )$ F $\quad 4 \times \sin ( 180 \times 4 \div \pi )$

Given that
$$\int _ { \log _ { 2 } 5 } ^ { \log _ { 2 } 20 } x \mathrm {~d} x = \log _ { 2 } M$$
what is the value of $M$ ?

$$\frac{1}{\log_{2} 6} + \frac{1}{\log_{3} 6}$$
Which of the following is this expression equal to?
A) $\frac{1}{3}$
B) $1$
C) $2$
D) $\log_{6} 2$
E) $\log_{6} 3$

For positive real numbers $a$, $b$, $c$ different from 1, $$\log_{a} b = \frac{1}{2}, \quad \log_{a} c = 3$$ Given this, what is the value of the expression $\log_{b}\left(\frac{b^{2}}{c\sqrt{a}}\right)$?
A) $\frac{3}{2}$
B) $\frac{5}{2}$
C) $\frac{5}{3}$
D) $-6$
E) $-5$

$12^{a} = 2$
$$6^{b} = 3$$
Given that, what is the value of the expression $\mathbf{12}^{\boldsymbol{(}\mathbf{1} - \mathbf{a}\mathbf{)2b}}$?
A) 15 B) 16 C) 9 D) 8 E) 4

Let $\mathbf { x }$ and $\mathbf { y }$ be real numbers.
$$2 ^ { x } - 2 ^ { -y } \left( 2 ^ { x+y } - 2 \right)$$
Which of the following is this expression equal to?
A) $2 ^ { x+1 }$
B) $2 ^ { y-x }$
C) $2 ^ { -y+1 }$
D) $\frac { 2 } { 9 }$
E) $\frac { 4 } { 9 }$

$$\log _ { 2 } \sqrt { 8 \sqrt { 4 \sqrt { 2 } } }$$
What is the result of this operation?
A) $\frac { 13 } { 8 }$
B) $\frac { 15 } { 8 }$
C) $\frac { 17 } { 8 }$
D) $\frac { 23 } { 16 }$
E) $\frac { 27 } { 16 }$

$\frac { \log _ { 3 } \sqrt { 27 } + \log _ { 27 } \sqrt { 3 } } { \log _ { 3 } \sqrt { 27 } - \log _ { 27 } \sqrt { 3 } }$\ What is the result of this operation?\ A) $\frac { 3 } { 2 }$\ B) $\frac { 4 } { 3 }$\ C) $\frac { 5 } { 4 }$\ D) $\frac { 6 } { 5 }$\ E) $\frac { 7 } { 6 }$

On a ruler-like scale with integers from 1 to 50 written on it, the distance of each integer $n$ from 1 is $\log n$ units.
When two identical rulers with this property are placed one below the other as shown in the figure, the number 42 on the upper ruler aligns with the number 28 on the lower ruler, and the number 33 on the upper ruler aligns with the number $x$ on the lower ruler.
Accordingly, what is $x$?
A) 18 B) 19 C) 20 D) 21 E) 22