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. $a < b < c$

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 \_\_\_\_

If real numbers $x, y, z$ satisfy $2 + \log_2 x = 3 + \log_3 y = 5 + \log_5 z$, then the size relationship of $x, y, z$ that is impossible is
A. $x > y > z$
B. $x > z > y$
C. $y > x > z$
D. $y > z > x$

If real numbers $x, y, z$ satisfy $2 + \log_2 x = 3 + \log_3 y = 5 + \log_5 z$, then the size relationship of $x, y, z$ that is impossible is
A. $x > y > z$
B. $x > z > y$
C. $y > x > z$
D. $y > z > x$

For each complex number $w$, we denote by $\operatorname{Re}(w)$ the real part of $w$. Show that, for all $z \in \stackrel{\circ}{\mathbb{D}}$: $$\ln|1-z| = -\operatorname{Re}\left(\sum_{n=1}^{\infty} \frac{z^n}{n}\right)$$ To do this, one may write $z = re^{i\theta}$ with $0 \leq r < 1$ and $\theta \in \mathbb{R}$, then study the function: $$\begin{aligned} F : [0,1[ &\rightarrow \mathbb{R} \\ \rho &\mapsto \ln\left|1 - \rho e^{i\theta}\right| \end{aligned}$$

Let $z \in \stackrel{\circ}{\mathbb{D}}$. Using the result that for all $z \in \stackrel{\circ}{\mathbb{D}}$, $\ln|1-z| = -\operatorname{Re}\left(\sum_{n=1}^{\infty}\frac{z^n}{n}\right)$, deduce that the Mahler measure of the polynomial $X - z$ is 1 and, in the case where $z \neq 0$, that of the polynomial $X - z^{-1}$ is $|z|^{-1}$.

Using the result of question 2.14, show that, for all $z \in \partial\mathbb{D}$, the Mahler measure of $X - z$ is 1.
To do this, one may be interested in the function: $$\begin{aligned} g : [0,1[ &\rightarrow \mathbb{R} \\ r &\mapsto M(X - rz) \end{aligned}$$ and note that, for all $r \in [0,1[$ and $\theta, \psi \in \mathbb{R}$, we have the inequality $\left|e^{i\theta} - re^{i\psi}\right| \geq |\sin(\theta - \psi)|$.

Let $\lambda$ be the leading coefficient of $Q$ and let $\alpha_1, \ldots, \alpha_n$ be the roots of $Q$ counted with multiplicity. Deduce from the previous questions that: $$M(Q) = |\lambda| \prod_{i=1}^n \max\left\{1, \left|\alpha_i\right|\right\}$$

Let $z \in D$. Show the convergence of the series $\sum_{n \geq 1} \frac{z^n}{n}$. Specify the value of its sum when $z \in ]-1,1[$. We denote $$L(z) := \sum_{n=1}^{+\infty} \frac{z^n}{n}$$

102- Suppose $\log_5(3x-2) = 1$. Given $\begin{vmatrix} \log 5 & \log 7 \\ \log 7 & \log 5 \end{vmatrix}$, what is the value of $x$?
(1) $9$ (2) $\dfrac{17}{7}$ (3) $4$ (4) $\dfrac{7}{3}$

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) $4$

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$ then $x$ equals
(A) $4 ^ { 10 }$
(B) 100
(C) $\log _ { 10 } 4$
(D) none of the above

If $\log_{12} 18 = a$, then $\log_{24} 16$ equals
(A) $\dfrac{8 - 4a}{5 - a}$ (B) $\dfrac{4a - 8}{a - 5}$ (C) $\dfrac{4 + 8a}{5 + a}$ (D) None of these

The last digit of $( 2004 ) ^ { 5 }$ is:
(a) 4
(b) 8
(c) 6
(d) 2

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$, then $x$ equals
(a) $4 ^ { 10 }$
(b) 100
(c) $\log _ { 10 } 4$
(d) none of the above.

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$, then $x$ equals
(a) $4 ^ { 10 }$
(b) 100
(c) $\log _ { 10 } 4$
(d) none of the above.

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$ then $x$ equals
(A) $4 ^ { 10 }$
(B) 100
(C) $\log _ { 10 } 4$
(D) none of the above

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$ then $x$ equals
(A) $4 ^ { 10 }$
(B) 100
(C) $\log _ { 10 } 4$
(D) none of the above

Let $a, b$ and $c$ be real numbers, each greater than 1, such that $$\frac{2}{3}\log_b a + \frac{3}{5}\log_c b + \frac{5}{2}\log_a c = 3$$ If the value of $b$ is 9, then the value of $a$ must be
(A) $\sqrt[3]{81}$
(B) $\frac{27}{2}$
(C) 18
(D) 27.

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 sum of all the solutions of $2 + \log _ { 2 } ( x - 2 ) = \log _ { ( x - 2 ) } 8$ in the interval $( 2 , \infty )$ is
(A) $\frac { 35 } { 8 }$.
(B) 5 .
(C) $\frac { 49 } { 8 }$.
(D) $\frac { 55 } { 8 }$.

28. The number of solutions of $\log 4 ( x - 1 ) = \log 2 ( x - 3 )$ is :
(A) 3
(B) 1
(C) 2
(D) 0

47. Let $\left( x _ { 0 } , y _ { 0 } \right)$ be the solution of the following equations
$$\begin{aligned} ( 2 x ) ^ { \ln 2 } & = ( 3 y ) ^ { \ln 3 } \\ 3 ^ { \ln x } & = 2 ^ { \ln y } . \end{aligned}$$
Then $x _ { 0 }$ is
(A) $\frac { 1 } { 6 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 1 } { 2 }$
(D) 6

ANSWER: C

1. The value of $\int _ { \sqrt { \ln 2 } } ^ { \sqrt { \ln 3 } } \frac { x \sin x ^ { 2 } } { \sin x ^ { 2 } + \sin \left( \ln 6 - x ^ { 2 } \right) } d x$ is
   (A) $\frac { 1 } { 4 } \ln \frac { 3 } { 2 }$
   (B) $\frac { 1 } { 2 } \ln \frac { 3 } { 2 }$
   (C) $\ln \frac { 3 } { 2 }$
   (D) $\frac { 1 } { 6 } \ln \frac { 3 } { 2 }$

ANSWER: A