We are to find the range of the values of $k$ such that the inequality
$$\frac { \log 3 x } { 4 x + 1 } \leqq \log \left( \frac { 2 k x } { 4 x + 1 } \right) \tag{1}$$
holds for all positive real numbers $x$, where $\log$ is the natural logarithm.
(1) For $\mathbf{A}$ and $\mathbf{B}$ in the following sentences, choose the correct answer from among (0) $\sim$ (8) below.
By transforming inequality (1) we obtain
$$\log k \geqq \mathbf { A } . \tag{2}$$
Here, when the right side of (2) is denoted by $g ( x )$ and this $g ( x )$ is differentiated with respect to $x$, we have
$$g ^ { \prime } ( x ) = \mathbf { B } .$$
(0) $\frac { \log 3 x } { 4 x + 1 } - \log ( 4 x + 1 ) - \log 2 x$
(1) $\frac { \log 3 x } { 4 x + 1 } - \log ( 4 x + 1 ) + \log 2 x$
(2) $\frac { \log 3 x } { 4 x + 1 } + \log ( 4 x + 1 ) + \log 2 x$
(3) $\frac { \log 3 x } { 4 x + 1 } + \log ( 4 x + 1 ) - \log 2 x$
(4) $\frac { 4 \log 3 x } { ( 4 x + 1 ) ^ { 2 } }$
(5) $\frac { 3 x + 2 + \log 3 x } { ( 4 x + 1 ) ^ { 2 } }$ (6) $- \frac { 4 \log 3 x } { ( 4 x + 1 ) ^ { 2 } }$ (7) $\frac { 3 x - 2 - \log 2 x } { ( 4 x + 1 ) ^ { 2 } }$ (8) $- \frac { 3 \log 2 x } { ( 4 x + 1 ) ^ { 2 } }$
(2) In the following sentences, for $\mathbf { E } , \mathbf { F }$ and $\mathbf { G }$, choose the correct answer from among (0) $\sim$ (3) below. For the other blanks, enter the correct number.
Over the interval $0 < x < \frac { \mathbf { C } } { \mathbf{D} }$, $g ( x )$ is $\mathbf { E }$ and over the interval $\frac { \mathbf{C} } { \mathbf{D} } < x$, $g ( x )$ is $\mathbf { F }$. Hence at $x = \frac { \mathbf { C } } { \mathbf{D} }$, $g ( x )$ is $\mathbf { G }$.
From the above, the range of the value of $k$ such that inequality (1) holds for all positive real numbers $x$ is
$$k \geqq \frac { \mathbf { H } } { \mathbf { I } }$$
(0) increasing
(1) decreasing
(2) maximized
(3) minimized

(Course 2) Answer the following questions, where log is the natural logarithm.
(1) Let $f ( x ) = x - 1 - \log x$. We are to find the minimum value of $f ( x )$.
First, we have
$$f ^ { \prime } ( x ) = \mathbf { A } - \frac { \mathbf { B } } { x } .$$
Examining the increases and decreases of the value of $f ( x )$, we see that at $x = \mathbf { C }$ the function is minimized and its value is $\mathbf { D }$. From this, we derive the inequality $x - 1 \geqq \log x$.
(2) For $\mathbf { G }$ in the following sentences, choose the correct answer from among choices (0) $\sim$ (3) below. For the other $\square$, enter the correct number.
Let $k$ be a positive real number and $n$ be a positive integer. We denote by $S$ the area of the figure bounded by the three straight lines $y = \frac { x } { n }$, $x = k$ and $y = 0$, and by $T$ the area of the figure bounded by the curve $y = \log x$, the straight line $x = k$ and the $x$ axis.

2. For ALL APPLICANTS.

In this question you may use without proof the following fact:
$$\ln ( 1 - x ) = - x - \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } \cdots - \frac { x ^ { n } } { n } \ldots \quad \text { for any } x \text { with } | x | < 1 .$$
[Note that $\ln x$ is alternative notation for $\log _ { e } x$.]
(i) By choosing a particular value of $x$ with $| x | < 1$, show that
$$\ln 2 = \frac { 1 } { 2 } + \frac { 1 } { 2 \times 2 ^ { 2 } } + \frac { 1 } { 3 \times 2 ^ { 3 } } + \frac { 1 } { 4 \times 2 ^ { 4 } } + \frac { 1 } { 5 \times 2 ^ { 5 } } + \ldots$$
(ii) Use part (i) and the fact that
$$\frac { 1 } { n 2 ^ { n } } < \frac { 1 } { 3 \times 2 ^ { n } } \quad \text { for } n \geqslant 4$$
to find the integer $k$ such that $\frac { k } { 24 } < \ln 2 < \frac { k + 1 } { 24 }$.
(iii) Show that
$$\ln \left( \frac { 3 } { 2 } \right) = \frac { 1 } { 2 } - \frac { 1 } { 2 \times 2 ^ { 2 } } + \frac { 1 } { 3 \times 2 ^ { 3 } } - \frac { 1 } { 4 \times 2 ^ { 4 } } + \frac { 1 } { 5 \times 2 ^ { 5 } } - \ldots$$
and deduce that
$$\ln 3 = 1 + \frac { 1 } { 3 \times 2 ^ { 2 } } + \frac { 1 } { 5 \times 2 ^ { 4 } } + \frac { 1 } { 7 \times 2 ^ { 6 } } + \ldots$$
(iv) Deduce that $\frac { 13 } { 12 } < \ln 3 < \frac { 11 } { 10 }$.
(v) Which is larger: $3 ^ { 17 }$ or $4 ^ { 13 }$ ? Without calculating either number, justify your answer.
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9. In a container, there are two types of bacteria, A and B. At any time, the product of the number of bacteria A and B is a constant $10 ^ { 10 }$. For simplicity, scientists use $P _ { A } = \log \left( n _ { A } \right)$ to record data about the number of bacteria A, where $n _ { A }$ is the number of bacteria A. Which of the following statements are correct?
(1) $1 \leq P _ { A } \leq 10$
(2) When $P _ { A } = 5$, the number of bacteria B equals the number of bacteria A
(3) If $P _ { A }$ measured last Monday was 4 and $P _ { A }$ measured last Friday was 8, then the number of bacteria A on Friday is twice the number on Monday
(4) If today's $P _ { A }$ value increases by 1 compared to yesterday, then today's bacteria A count is 10 more than yesterday's
(5) If the scientist controls the number of bacteria B to be 50,000, then $5 < P _ { A } < 5.5$

Let $a = \log _ { 2 } 8 , ~ b = \log _ { 3 } 1 , ~ c = \log _ { 0.5 } 8$. Select the correct options.
(1) $b = 0$
(2) $a + b + c > 0$
(3) $a > b > c$
(4) $a ^ { 2 } > b ^ { 2 } > c ^ { 2 }$
(5) $2 ^ { a } > 3 ^ { b } > \left( \frac { 1 } { 2 } \right) ^ { c }$

A certain brand of calculator computes the logarithm $\log _ { a } b$ by pressing $\log$（1）$a$（ $b$ ）. A student computing $\log _ { a } b$ (where $a > 1$ and $b > 1$ ) pressed the buttons in the wrong order, pressing $\log$（1）$b$（ $a$ ） instead, obtaining a result that is $\frac { 9 } { 4 }$ times the correct value. Select the relationship between $a$ and $b$.
(1) $a ^ { 2 } = b ^ { 3 }$
(2) $a ^ { 3 } = b ^ { 2 }$
(3) $a ^ { 4 } = b ^ { 9 }$
(4) $2 a = 3 b$
(5) $3 a = 2 b$

If $x, y$ are two positive real numbers satisfying $x^{-\frac{1}{3}} y^{2} = 1$ and $2\log y = 1$, then $\frac{x - y^{2}}{10} =$ (13--1) (13--2).

Two positive real numbers $a$ and $b$ satisfy $ab^{2} = 10^{5}$ and $a^{2}b = 10^{3}$. Then $\log b = \dfrac{\square}{\square}$. (Express as a fraction in lowest terms)

Given that $a , b , c$ are real numbers satisfying $1 < a < 10$, $b = \log a$, $c = \log b$, select the correct option.
(1) $c < 0 < b < 1$
(2) $0 < c < 1 < b$
(3) $0 < c < b < 1$
(4) $1 < c < b$
(5) $c < b < 0$

Given that real numbers $a , b$ satisfy $\frac { 1 } { 2 } < a < 1$ and $1 < b < 2$ . Which of the following options has the smallest value?
(1) 0
(2) $\log a$
(3) $\log \left( a ^ { 2 } \right)$
(4) $\log b$
(5) $\frac { 1 } { \log b }$

On the coordinate plane, a point whose $x$-coordinate and $y$-coordinate are both integers is called a lattice point. How many lattice points are in the interior (not including the boundary) of the region bounded by the function graph $y = \log _ { 2 } x$, the $x$-axis, and the line $x = \frac { 61 } { 2 }$?
(1) 88
(2) 89
(3) 90
(4) 91
(5) 92

Select the value of $\sum _ { k = 1 } ^ { 5 } \log _ { 7 } \left( \frac { 2 k - 1 } { 2 k + 1 } \right)$.
(1) $- \log 11$
(2) $\log 11$
(3) $\log \frac { 11 } { 7 }$
(4) $- \frac { \log 11 } { \log 7 }$
(5) $\frac { \log 11 } { \log 7 }$

5. Given that $y = - \log _ { 10 } ( 1 - x )$ for $x < 1$, find $x$ in terms of $y$.
A $\quad x = - \frac { 1 } { \log _ { 10 } ( 1 - y ) }$
B $x = 1 + \log _ { 10 } y$
C $x = 1 - \log _ { 10 } y$
D $\quad x = 1 - 10 ^ { - y }$
E $\quad x = 10 ^ { - y } - 1$
F $\quad x = 10 ^ { 1 - y }$

5. Using the observation that $2 ^ { 5 } \approx 3 ^ { 3 }$, it is possible to deduce that $\log _ { 3 } 2$ is approximately
A $\frac { 3 } { 5 }$
B $\frac { 2 } { 3 }$
C $\quad \frac { 3 } { 2 }$
D $\frac { 5 } { 3 }$
E $\frac { 1 } { 2 }$
F 2

8. Given that $a ^ { x } b ^ { 2 x } c ^ { 3 x } = 2$, where $a , b$, and $c$ are positive real numbers, then $x =$
A $\quad \log _ { 10 } \left( \frac { 2 } { a + 2 b + 3 c } \right)$
B $\frac { \log _ { 10 } 2 } { \log _ { 10 } ( a + 2 b + 3 c ) }$
C $\quad \frac { 2 } { \log _ { 10 } ( a + 2 b + 3 c ) }$
D $\frac { 2 } { a + 2 b + 3 c }$
E $\quad \log _ { 10 } \left( \frac { 2 } { a b ^ { 2 } c ^ { 3 } } \right)$ F $\quad \frac { \log _ { 10 } 2 } { \log _ { 10 } \left( a b ^ { 2 } c ^ { 3 } \right) }$ G $\quad \frac { 2 } { \log _ { 10 } \left( a b ^ { 2 } c ^ { 3 } \right) }$ H $\frac { 2 } { a b ^ { 2 } c ^ { 3 } }$

10. Which one of the following is a sketch of the graph of $y = \log _ { x } 2$ for $x > 1$ ? \textbackslash begin\{tabular\} \{ | r | r l l l l l l l l | r l | r | \} \textbackslash hline $5 -$ $4 -$ $3 -$ $2 -$ $1 -$ 0

• 1 $- 2 -$ \textbackslash end\{tabular\}

1. Which one of the following numbers is largest in value? (All angles are given in radians.)

A $\tan \left( \frac { 3 \pi } { 4 } \right)$
B $\quad \log _ { 10 } 100$
C $\quad \sin ^ { 10 } \left( \frac { \pi } { 2 } \right)$
D $\quad \log _ { 2 } 10$
E $( \sqrt { 2 } - 1 ) ^ { 10 }$

The four real numbers $a , b , c$, and $d$ are all greater than 1 .
Suppose that they satisfy the equation $\log _ { c } d = \left( \log _ { a } b \right) ^ { 2 }$.
Use some of the lines given to construct a proof that, in this case, it follows that
$$( * ) \log _ { b } d = \left( \log _ { a } b \right) \left( \log _ { a } c \right)$$
(1) Let $x = \log _ { a } b$ and $y = \log _ { a } c$
(2) $d = \left( c ^ { x } \right) ^ { 2 }$
(3) $d = c ^ { \left( x ^ { 2 } \right) }$
(4) $d = b ^ { x y }$
(5) $d = \left( a ^ { y } \right) ^ { \left( x ^ { 2 } \right) }$
(6) $d = \left( \left( a ^ { y } \right) ^ { x } \right) ^ { 2 }$
(7) $d = \left( a ^ { x } \right) ^ { x y }$
(8) $d = a ^ { \left( y ^ { 2 x } \right) }$
(9) $d = a ^ { \left( x ^ { 2 } y \right) }$

The real roots of the equation $4 ^ { 2 x } + 12 = 2 ^ { 2 x + 3 }$ are $p$ and $q$, where $p > q$. The value of $p - q$ can be expressed as
A $\frac { 3 } { 4 }$ B 1 C 4 D $- \frac { 1 } { 2 } + \log _ { 10 } \frac { 3 } { 2 }$ E $\frac { \log _ { 10 } 3 } { \log _ { 10 } 4 }$ F $\frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$

Given the simultaneous equations
$$\begin{gathered} \log _ { 10 } 2 + \log _ { 10 } ( y - 1 ) = 2 \log _ { 10 } x \\ \log _ { 10 } ( y + 3 - 3 x ) = 0 \end{gathered}$$
the values of $y$ are
A $\frac { 5 } { 2 } \pm \frac { 3 \sqrt { 5 } } { 2 }$ B $3 \pm \sqrt { 3 }$ C $7 \pm 3 \sqrt { 3 }$ D 3,9 E 1,13

Which one of the following numbers is smallest in value?
A $\quad \log _ { 2 } 7$
B $\left( 2 ^ { - 3 } + 2 ^ { - 2 } \right) ^ { - 1 }$
C $2 ^ { ( \pi / 3 ) }$
D $\frac { 1 } { 4 ( \sqrt { 2 } - 1 ) ^ { 3 } }$
E $\quad 4 \sin ^ { 2 } \left( \frac { \pi } { 4 } \right)$

Consider the following problem:
Solve the inequality $\left( \frac { 1 } { 4 } \right) ^ { n } < \left( \frac { 1 } { 32 } \right) ^ { 10 }$, where $n$ is a positive integer.
A student produces the following argument:
$$\begin{array} { r l r } \left( \frac { 1 } { 4 } \right) ^ { n } & < \left( \frac { 1 } { 32 } \right) ^ { 10 } & \\ \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 4 } \right) ^ { n } & < \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 32 } \right) ^ { 10 } & ( \mathrm { I } ) \\ n \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 4 } \right) & < 10 \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 32 } \right) & \downarrow ( \mathrm { II } ) \\ n < \frac { 10 \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 32 } \right) } { \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 4 } \right) } & \downarrow ( \mathrm { III } ) \\ n < \frac { 10 \times 5 } { 2 } = 25 & \downarrow ( \mathrm { IV } ) \\ 1 \leqslant n \leqslant 24 & \downarrow ( \mathrm {~V} ) \end{array}$$
Which step (if any) in the argument is invalid?
A There are no invalid steps; the argument is correct
B Only step (I) is invalid; the rest are correct
C Only step (II) is invalid; the rest are correct
D Only step (III) is invalid; the rest are correct
E Only step (IV) is invalid; the rest are correct
F Only step (V) is invalid; the rest are correct

The line $y = m x + 4$ passes through the points ( $3 , \log _ { 2 } p$ ) and ( $\log _ { 2 } p , 4$ ). What are the possible values of $p$ ?
A $p = 1$ and $p = 4$
B $p = 1$ and $p = 16$
C $\quad p = \frac { 1 } { 4 } \quad$ and $\quad p = 4$
D $\quad p = \frac { 1 } { 4 } \quad$ and $\quad p = 64$
E $\quad p = \frac { 1 } { 64 }$ and $p = 4$
F $\quad p = \frac { 1 } { 64 }$ and $p = 16$

Find the sum of the real values of $x$ that satisfy the simultaneous equations:
$$\begin{aligned} \log_3(xy^2) &= 1 \\ (\log_3 x)(\log_3 y) &= -3 \end{aligned}$$

Find the real non-zero solution to the equation
$$\frac{2^{(9^x)}}{8^{(3^x)}} = \frac{1}{4}$$

The numbers $a , b$ and $c$ are each greater than 1 .
The following logarithms are all to the same base:
$$\begin{aligned} \log \left( a b ^ { 2 } c \right) & = 7 \\ \log \left( a ^ { 2 } b c ^ { 2 } \right) & = 11 \\ \log \left( a ^ { 2 } b ^ { 2 } c ^ { 3 } \right) & = 15 \end{aligned}$$
What is this base?