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) $\frac { 7 } { 8 }$

The domain of the function $f ( \mathrm { x } ) = \log _ { 2 } \left( \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 3 \right)$ is
(A) $[ - 3,1 ]$
(B) $( - 3,1 )$
(C) $( - \infty , - 3 ] \cup [ 1 , + \infty )$
(D) $( - \infty , - 3 ) \cup ( 1 , + \infty )$

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

8. Let real numbers $a , b , t$ satisfy $| a + 1 | = | \sin b | = t$
[Figure]
(Figure for Question 7)
A. If $t$ is determined, then $b ^ { 2 }$ is uniquely determined
B. If $t$ is determined, then $a ^ { 2 } + 2 a$ is uniquely determined
C. If $t$ is determined, then $\sin \frac { b } { 2 }$ is uniquely determined
D. If $t$ is determined, then $a ^ { 2 } + a$ is uniquely determined

II. Fill-in-the-Blank Questions (This section contains 7 questions. Multi-blank questions are worth 6 points each, single-blank questions are worth 4 points each, 36 points total.)

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

The monotone increasing interval of the function $f(x) = \ln(x^2 - 2x - 9)$ is
A. $(-\infty, -2)$
B. $(-\infty, 1)$
C. $(1, +\infty)$
D. $(4, +\infty)$

Given the function $f(x) = \ln x + \ln(2 - x)$, then
A. $f(x)$ is monotonically increasing on $(0, 2)$
B. $f(x)$ is monotonically decreasing on $(0, 2)$
C. $f(x)$ is increasing on $(0, 1)$ and decreasing on $(1, 2)$
D. $f(x)$ is decreasing on $(0, 1)$ and increasing on $(1, 2)$

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) $10 ^ { - 10.1 }$

Let $f ( x )$ be an even function with domain $\mathbf { R }$ that is monotonically decreasing on $( 0 , + \infty )$. Then
A. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right)$
B. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right)$
C. $f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$
D. $f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$

14. Given that $f ( x )$ is an odd function, and when $x < 0$, $f ( x ) = - \mathrm { e } ^ { a x }$. If $f ( \ln 2 ) = 8$, then $a =$ $\_\_\_\_$ .

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

6. Vision of adolescents 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 $L = 5 + \lg V$. It is known that a student's vision data in the five-point recording method is 4.9. Then the student's 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

7. Given $a = \log _ { 5 } 2 , b = \log _ { 8 } 3 , c = \frac { 1 } { 2 }$, which of the following judgments is correct? ( )
A. $c < b < a$
B. $b < a < c$
C. $a < c < b$
D. $a < b < c$
【Answer】C 【Solution】 【Analysis】Use the monotonicity of logarithmic functions to compare the sizes of $a$, $b$, and $c$, and thus reach the conclusion. 【Detailed Solution】 $a = \log _ { 5 } 2 < \log _ { 5 } \sqrt { 5 } = \frac { 1 } { 2 } = \log _ { 8 } 2 \sqrt { 2 } < \log _ { 8 } 3 = b$, that is, $a < c < b$. Therefore, the answer is: C.

Given $a = \log _ { 5 } 2 , b = \log _ { 8 } 3 , c = \frac { 1 } { 2 }$, which of the following judgments is correct?
A. $c < b < a$
B. $b < a < c$
C. $a < c < b$
D. $a < b < c$

7. Let $a = 0.1 \mathrm { e } ^ { 0.1 }$, $b = \frac { 1 } { 9 }$, $c = - \ln 0.9$. Then
A. $a < b < c$
B. $c < b < a$
C. $c < a < b$
D. $a < c < b$

Let the water quality index be $d = \frac { S - 1 } { \ln n }$, and the larger $d$ is, the better the water quality. If $S$ remains constant and $d _ { 1 } = 2.1 , d _ { 2 } = 2.2$, then the relationship between $n _ { 1 }$ and $n_2$ is \_\_\_\_

Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Deduce that
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \frac { \pi } { \sin ( \pi x ) }$$