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

QUESTION 161
The value of $e^0 + \ln 1$ is
(A) 0
(B) 1
(C) 2
(D) $e$
(E) $e + 1$

To take the trip of her dreams, a person needed to take out a loan in the amount of $\mathrm{R}\$ 5000.00$. To pay the installments, she has at most $\mathrm{R}\$ 400.00$ monthly. For this loan amount, the installment value ($P$) is calculated as a function of the number of installments ($n$) according to the formula
$$P = \frac { 5000 \times 1.013 ^ { n } \times 0.013 } { \left( 1.013 ^ { n } - 1 \right) }$$
If necessary, use 0.005 as an approximation for $\log 1.013$; 2.602 as an approximation for $\log 400$; 2.525 as an approximation for $\log 335$.
According to the given formula, the smallest number of installments whose values do not compromise the limit defined by the person is
(A) 12.
(B) 14.
(C) 15.
(D) 16.
(E) 17.

The equation $\log_2(x+1) = 3$ has solution:
(A) $x = 6$
(B) $x = 7$
(C) $x = 8$
(D) $x = 9$
(E) $x = 10$

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

An entrepreneur uses machines whose internal pressure $P$, in atmosphere, depends on the continuous time of use $t$, in hour, and on a positive parameter $K$, which defines the model of the machine, according to the expression: $$P = 4 \cdot \log[-K \cdot (t + 1) \cdot (t - 19)]$$
The manufacturer of these machines recommends to the user that the internal pressure of this type of machine does not exceed 10 atmospheres during its operation.
The entrepreneur intends to buy new machines of this type that should operate, daily, for a continuous period of 10 hours. For this, he needs to define the model of machine to be acquired by choosing the largest possible value of the parameter $K$, in accordance with the manufacturer's recommendation. The largest value to be chosen for $K$ is
(A) $10^{0.5}$
(B) $10^{8}$
(C) $\dfrac{10^{2.5}}{84}$
(D) $\dfrac{10^{2.5}}{99}$
(E) $25 \times 10^{-2}$

If $a , b , c$ are real numbers $> 1$, then show that $$\frac { 1 } { 1 + \log _ { a ^ { 2 } b } \frac { c } { a } } + \frac { 1 } { 1 + \log _ { b ^ { 2 } c } \frac { a } { b } } + \frac { 1 } { 1 + \log _ { c ^ { 2 } a } \frac { b } { c } } = 3$$

Consider the two equations numbered [1] and [2]:
$$\begin{aligned} \log _ { 2021 } a & = 2022 - a \\ 2021 ^ { b } & = 2022 - b \end{aligned}$$
(a) Equation [1] has a unique solution.
(b) Equation [2] has a unique solution.
(c) There exists a solution $a$ for [1] and a solution $b$ for [2] such that $a = b$.
(d) There exists a solution $a$ for [1] and a solution $b$ for [2] such that $a + b$ is an integer.

Statements
(17) Let $a = \frac{1}{\ln 3}$. Then $3^a = e$. (18) $\sin(0.02) < 2\sin(0.01)$. (19) $\arctan(0.01) > 0.01$. (20) $4\int_0^1 \arctan(x)\, dx = \pi - \ln 4$.

Statements
(17) $\sqrt [ 4 ] { 4 } < \sqrt [ 5 ] { 5 }$. (18) $\log _ { 10 } 11 > \log _ { 11 } 12$. (19) $\frac { \pi } { 4 } < \sqrt { 2 - \sqrt { 2 } }$. (20) $( 2022 ! ) ^ { 2 } > 2022 ^ { 2022 }$.

14. Let $x , y , z$ be positive numbers such that $x ^ { 2 } + y ^ { 2 } = z ^ { 2 }$. Determine the value of the following expression:
$$\frac { \log _ { y + z } x + \log _ { z - y } x } { \left( \log _ { y + z } x \right) \left( \log _ { z - y } x \right) }$$
(a) Undefined.
(b) $\frac { 1 } { 2 }$
(c) $\frac { 1 } { 4 }$
(d) 2

From the following , select all correct statements. [3 points]
ㄱ. $2 ^ { \log _ { 2 } 1 + \log _ { 2 } 2 + \log _ { 2 } 3 + \cdots + \log _ { 2 } 10 } = 10$ ㄴ. $\log _ { 2 } \left( 2 ^ { 1 } \times 2 ^ { 2 } \times 2 ^ { 3 } \times \cdots \times 2 ^ { 10 } \right) ^ { 2 } = 55 ^ { 2 }$ ㄷ. $\left( \log _ { 2 } 2 ^ { 1 } \right) \left( \log _ { 2 } 2 ^ { 2 } \right) \left( \log _ { 2 } 2 ^ { 3 } \right) \cdots \left( \log _ { 2 } 2 ^ { 10 } \right) = 55$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Which of the following in $\langle$Remarks$\rangle$ are correct? [3 points]
$\langle$Remarks$\rangle$ ㄱ. $2 ^ { \log _ { 2 } 1 + \log _ { 2 } 2 + \log _ { 2 } 3 + \cdots + \log _ { 2 } 10 } = 10 !$ ㄴ. $\log _ { 2 } \left( 2 ^ { 1 } \times 2 ^ { 2 } \times 2 ^ { 3 } \times \cdots \times 2 ^ { 10 } \right) ^ { 2 } = 55 ^ { 2 }$ ㄷ. $\left( \log _ { 2 } 2 ^ { 1 } \right) \left( \log _ { 2 } 2 ^ { 2 } \right) \left( \log _ { 2 } 2 ^ { 3 } \right) \cdots \left( \log _ { 2 } 2 ^ { 10 } \right) = 55$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

System of inequalities $$\left\{ \begin{array} { l } \log _ { 3 } | x - 3 | < 4 \\ \log _ { 2 } x + \log _ { 2 } ( x - 2 ) \geqq 3 \end{array} \right.$$ Find the number of integers $x$ that satisfy the system. [3 points]

Let $a$ be the largest integer among numbers whose common logarithm characteristic is 2, and let $b$ be the smallest number among numbers whose common logarithm characteristic is $-2$. What is the value of $ab$? [4 points]
(1) 0.9
(2) 0.99
(3) 1
(4) 9.99
(5) 10

For a positive number $a$, let the characteristic and mantissa of $\log a$ be $f ( a )$ and $g ( a )$ respectively. Which of the following in are correct? [3 points] 〈Remarks〉 ㄱ. $f ( 2006 ) = 3$ ㄴ. $g ( 2 ) + g ( 6 ) = g ( 12 ) + 1$ ㄷ. If $f ( a b ) = f ( a ) + f ( b )$, then $g ( a b ) = g ( a ) + g ( b )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a positive number $a$, let the characteristic and mantissa of $\log a$ be $f ( a )$ and $g ( a )$ respectively. Which of the following statements in are true? [3 points]

ㄱ. $f ( 2006 ) = 3$ ㄴ. $g ( 2 ) + g ( 6 ) = g ( 12 ) + 1$ ㄷ. If $f ( a b ) = f ( a ) + f ( b )$, then $g ( a b ) = g ( a ) + g ( b )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For two positive numbers $a , b$, $$\left\{ \begin{array} { l } ab = 27 \\ \log _ { 3 } \frac { b } { a } = 5 \end{array} \right.$$ When these conditions hold, find the value of $4 \log _ { 3 } a + 9 \log _ { 3 } b$. [3 points]

For the function with domain $\{ x \mid 1 \leqq x \leqq 81 \}$, $$y = \left( \log _ { 3 } x \right) \left( \log _ { \frac { 1 } { 3 } } x \right) + 2 \log _ { 3 } x + 10$$ Let $M$ be the maximum value and $m$ be the minimum value. Find the value of $M + m$. [4 points]

To remove bacteria living in a water tank, a chemical is to be administered. Let $C _ { 0 }$ be the initial number of bacteria per 1 mL of water in the tank, and let $C$ be the number of bacteria per 1 mL at time $t$ hours after the chemical is administered. The following relationship holds: $$\log \frac { C } { C _ { 0 } } = - k t \quad ( k \text { is a positive constant } )$$ The initial number of bacteria per 1 mL of water is $8 \times 10 ^ { 5 }$, and at time 3 hours after the chemical is administered, the number of bacteria per 1 mL becomes $2 \times 10 ^ { 5 }$. After $a$ hours from administering the chemical, the number of bacteria per 1 mL first becomes $8 \times 10 ^ { 3 }$ or less. Find the value of $a$. (Here, calculate using $\log 2 = 0.3$.) [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

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

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

For an integer $n$, two sets $A ( n ) , B ( n )$ are defined as $$\begin{aligned} & A ( n ) = \left\{ x \mid \log _ { 2 } x \leqq n \right\} \\ & B ( n ) = \left\{ x \mid \log _ { 4 } x \leqq n \right\} \end{aligned}$$ Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $A ( 1 ) = \{ x \mid 0 < x \leqq 1 \}$ ㄴ. $A ( 4 ) = B ( 2 )$ ㄷ. When $A ( n ) \subset B ( n )$, then $B ( - n ) \subset A ( - n )$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ

For an integer $n$, two sets $A ( n ) , B ( n )$ are defined as $$\begin{aligned} & A ( n ) = \left\{ x \mid \log _ { 2 } x \leqq n \right\} \\ & B ( n ) = \left\{ x \mid \log _ { 4 } x \leqq n \right\} \end{aligned}$$ Which of the following statements in are correct? [4 points]
Remarks ㄱ. $A ( 1 ) = \{ x \mid 0 < x \leqq 1 \}$ ㄴ. $A ( 4 ) = B ( 2 )$ ㄷ. When $A ( n ) \subset B ( n )$, we have $B ( - n ) \subset A ( - n )$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ