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 $$L(z) := \sum_{n=1}^{+\infty} \frac{z^n}{n}$$
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.