$\lim _ { h \rightarrow 0 } \frac { \ln ( 4 + h ) - \ln ( 4 ) } { h }$ is
(A) 0
(B) $\frac { 1 } { 4 }$
(C) 1
(D) $e$
(E) nonexistent

Exercise 4 (7 points) Theme: natural logarithm function, probabilities This exercise is a multiple choice questionnaire (MCQ) comprising six questions. The six questions are independent. For each question, only one of the four answers is correct. The candidate will indicate on his answer sheet the number of the question followed by the letter corresponding to the correct answer. No justification is required.
A wrong answer, a multiple answer or no answer gives neither points nor deducts any points.
Question 1 The real number $a$ defined by $a = \ln ( 9 ) + \ln \left( \frac { \sqrt { 3 } } { 3 } \right) + \ln \left( \frac { 1 } { 9 } \right)$ is equal to: a. $1 - \frac { 1 } { 2 } \ln ( 3 )$ b. $\frac { 1 } { 2 } \ln ( 3 )$ c. $3 \ln ( 3 ) + \frac { 1 } { 2 }$ d. $- \frac { 1 } { 2 } \ln ( 3 )$
Question 2 We denote by $(E)$ the following equation $\ln x + \ln ( x - 10 ) = \ln 3 + \ln 7$ with unknown real $x$. a. 3 is a solution of $(E)$. b. $5 - \sqrt { 46 }$ is a solution of $(E)$. c. The equation $(E)$ admits a unique real solution. d. The equation $(E)$ admits two real solutions.
Question 3 The function $f$ is defined on the interval $] 0 ; + \infty [$ by the expression $f ( x ) = x ^ { 2 } ( - 1 + \ln x )$. We denote by $\mathscr { C } _ { f }$ its representative curve in the plane with a coordinate system. a. For every real $x$ in the interval $] 0 ; + \infty [$ , $f ^ { \prime } ( x ) = 2 x + \frac { 1 } { x }$. b. The function $f$ is increasing on the interval $] 0 ; + \infty [$. c. $f ^ { \prime } ( \sqrt { \mathrm { e } } )$ is different from 0. d. The line with equation $y = - \frac { 1 } { 2 } e$ is tangent to the curve $\mathscr { C } _ { f }$ at the point with abscissa $\sqrt { e }$.
Question 4
A bag contains 20 yellow tokens and 30 blue tokens. We draw successively and with replacement 5 tokens from the bag. The probability of drawing exactly 2 yellow tokens, rounded to the nearest thousandth, is: a. 0.683 b. 0.346 c. 0.230 d. 0.165
Question 5
A bag contains 20 yellow tokens and 30 blue tokens. We draw successively and with replacement 5 tokens from the bag. The probability of drawing at least one yellow token, rounded to the nearest thousandth, is: a. 0.078 b. 0.259 c. 0.337 d. 0.922
Question 6
A bag contains 20 yellow tokens and 30 blue tokens. We perform the following random experiment: we draw successively and with replacement five tokens from the bag. We note the number of yellow tokens obtained after these five draws. If we repeat this random experiment a very large number of times then, on average, the number of yellow tokens is equal to: a. 0.4 b. 1.2 c. 2 d. 2.5

Question 170
O logaritmo de 1 000 na base 10 é
(A) 1 (B) 2 (C) 3 (D) 4 (E) 10

O valor de $\log_{10} 1000 + \log_{10} 0{,}01$ é
(A) $-1$ (B) $0$ (C) $1$ (D) $2$ (E) $3$

QUESTION 149
The value of $\log_2 32$ is
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

The value of $\log_3 81$ is:
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

For a real number $a$ ($a > 1$), let $b = \sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { a } \right) ^ { n }$ be represented as in [Figure 1], and for a real number $c$, let $d = 16 ^ { c }$ be represented as in [Figure 2].
For the real numbers $x$, $y$, $z$ in the figure below, find the value of $\frac { x z } { y }$. [4 points]

What is the value of $\left( \log _ { 3 } 27 \right) \times 8 ^ { \frac { 1 } { 3 } }$? [2 points]
(1) 12
(2) 10
(3) 8
(4) 6
(5) 4

For three real numbers $a , b , c$ greater than 1, when $\log _ { a } c : \log _ { b } c = 2 : 1$, what is the value of $\log _ { a } b + \log _ { b } a$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

The value of $\left( \log _ { 3 } 27 \right) \times 8 ^ { \frac { 1 } { 3 } }$ is? [2 points]
(1) 12
(2) 10
(3) 8
(4) 6
(5) 4

What is the value of $8 ^ { \frac { 2 } { 3 } } + \log _ { 2 } 8$? [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

The value of $8 ^ { \frac { 2 } { 3 } } + \log _ { 2 } 8$ is? [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

When $a = \log _ { 2 } 10 , b = 2 \sqrt { 2 }$, what is the value of $a \log b$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

What is the value of $4 ^ { \frac { 3 } { 2 } } \times \log _ { 3 } \sqrt { 3 }$? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

The sequence $\{a_n\}$ satisfies the following for all natural numbers $n$: $$\sum_{k=1}^{n} a_k = \log \frac{(n+1)(n+2)}{2}$$ Let $\sum_{k=1}^{20} a_{2k} = p$. Find the value of $10^p$. [4 points]

What is the value of $4 ^ { \frac { 3 } { 2 } } \times \log _ { 3 } \sqrt { 3 }$? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

What is the value of $\log _ { 2 } 40 - \log _ { 2 } 5$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + x ) } { 3 x }$? [2 points]
(1) 1
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 1 } { 5 }$

What is the value of $\log _ { 15 } 3 + \log _ { 15 } 5$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + 5 x ) } { e ^ { 2 x } - 1 }$? [2 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

Find the value of $\log _ { 3 } 72 - \log _ { 3 } 8$. [3 points]

Find the number of natural numbers $n$ such that $\log _ { 4 } 2 n ^ { 2 } - \frac { 1 } { 2 } \log _ { 2 } \sqrt { n }$ is a natural number not exceeding 40. [4 points]

Find the value of $\log _ { 2 } 120 - \frac { 1 } { \log _ { 15 } 2 }$. [3 points]

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