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 $2 ^ { a } + 2 ^ { - a } =$ $\_\_\_\_$ .

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 $\_\_\_\_$.

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) $\frac { \log 11 } { \log 7 }$

$$\frac{1}{\log_{2} 6} + \frac{1}{\log_{3} 6}$$
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

Ada calculates the value of $\log _ { 2 } n$ on her scientific calculator for every positive integer n where $\mathrm { n } \leq 32$, and observes that each value is either an integer or a decimal number. Ada writes down either the number itself if the value displayed on the screen is an integer, or the integer part of the number if it is a decimal, and then finds the sum of these numbers she wrote down. Accordingly, what is the result of the sum that Ada found?
A) 94
B) 97
C) 100
D) 103
E) 106

On a calculator, when an operation is performed, the machine displays the result as a whole number if the result is an integer, or displays the integer part along with the first two decimal places after the decimal point if the result is a decimal number.
When Nevzat performs the operation $\ln ( 9{,}6 )$ on this calculator, he sees the value 2.26 on the screen, and when he performs the operation $\ln ( 0{,}3 )$, he sees the value $-1{,}20$ on the screen.
When Nevzat performs the operation $\ln ( 0{,}5 )$ on this calculator, what value does he see on the screen?
A) $-0{,}61$
B) $-0{,}65$
C) $-0{,}69$
D) $-0{,}73$
E) $-0{,}77$