If $\lim _ { \mathrm { t } \rightarrow 0 } \left( \int _ { 0 } ^ { 1 } ( 3 x + 5 ) ^ { \mathrm { t } } \mathrm { d } x \right) ^ { \frac { 1 } { t } } = \frac { \alpha } { 5 \mathrm { e } } \left( \frac { 8 } { 5 } \right) ^ { \frac { 2 } { 3 } }$, then $\alpha$ is equal to $\_\_\_\_$

Let $f ( x ) = \lim _ { \mathrm { n } \rightarrow \infty } \sum _ { \mathrm { r } = 0 } ^ { \mathrm { n } } \left( \frac { \tan \left( x / 2 ^ { r + 1 } \right) + \tan ^ { 3 } \left( x / 2 ^ { r + 1 } \right) } { 1 - \tan ^ { 2 } \left( x / 2 ^ { r + 1 } \right) } \right)$. Then $\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { f ( x ) } } { ( x - f ( x ) ) }$ is equal to

Given a sequence $\left\{ a _ { n } \right\}$ that satisfies the following conditions
$$\begin{aligned} & a _ { 1 } = 1 \\ & a _ { n + 1 } = 2 a _ { n } ^ { 2 } \quad ( n = 1,2,3 , \cdots ) , \end{aligned}$$
we are to find the number of natural numbers $n$ satisfying $a _ { n } < 10 ^ { 60 }$. (For the value of $\log _ { 10 } 2$, use the approximation 0.301.)
In this sequence we note that $a _ { n } > 0$ for all natural numbers $n$. Thus when we consider common logarithms of both sides of (1), we have
$$\log _ { 10 } a _ { n + 1 } = \log _ { 10 } \mathbf { A } + \mathbf { B } \log _ { 10 } a _ { n } .$$
When we set $b _ { n } = \log _ { 10 } a _ { n } + \log _ { 10 } \mathbf{A}$, the sequence $\left\{ b _ { n } \right\}$ is a geometric progression such that the common ratio is $\mathbf { C }$. Then
$$\log _ { 10 } a _ { n } = \left( ( \mathbf { D } ) ^ { n - 1 } - \mathbf { E } \right) \log _ { 10 } \mathbf { F } .$$
Furthermore, since $a _ { n } < 10 ^ { 60 }$,
$$\mathbf{D}^{ n - 1 } < \frac { \mathbf { G H } } { \log _ { 10 } \mathbf { F } } + \mathbf { E }$$
Since $\mathbf{IJK}$ is the least natural number which is larger than the value of the right side of (2), the number of natural numbers $n$ satisfying $a _ { n } < 10 ^ { 60 }$ is $\mathbf{L}$.

Let $x$ and $y$ be positive numbers which satisfy
$$\left(\log_2 x\right)^2 + \left(\log_2 y\right)^2 = \log_2 \frac{8x^2}{y^2}. \tag{1}$$
We are to find the maximum value of $xy^2$ and the values of $x$ and $y$ at that point.
(1) The right side of (1) can be transformed into
$$\log_2 \frac{8x^2}{y^2} = \mathbf{A}\log_2 x - \mathbf{B}\log_2 y + \mathbf{C}.$$
So, setting $X = \log_2 x$ and $Y = \log_2 y$, we can express (1) in terms of $X$ and $Y$ as
$$(X - \mathbf{D})^2 + (Y + \mathbf{E})^2 = \mathbf{F}.$$
(2) Set $k = \log_2 xy^2$. Using $X$ and $Y$ above, this equality can be transformed into
$$X + \mathbf{GG}\,Y - k = 0.$$
If we graph (2) and (3) on a plane with coordinates $(X, Y)$, the graph of (2) is a circle, and the graph of (3) is a straight line. When $k$ is maximized, the graph of (3) is tangent to the graph of (2). Hence, when $k = \mathbf{H}$, $xy^2$ takes the maximum value IJ and in this case $x = \mathbf{K}$ and $y = \mathbf{L}$.

Let $x$ satisfy the inequality
$$2 \left( \log _ { \frac { 1 } { 3 } } x \right) ^ { 2 } + 9 \log _ { \frac { 1 } { 3 } } x + 9 \leqq 0 .$$
We are to find the maximum value of the function
$$f ( x ) = \left( \log _ { 3 } x \right) \left( \log _ { 3 } \frac { x } { 3 } \right) \left( \log _ { 3 } \frac { x } { 9 } \right) .$$
The range of values of $x$ satisfying (1) is
$$\mathbf { A } \sqrt { \mathbf { B } } \leqq x \leqq \mathbf { C D } .$$
When we set $t = \log _ { 3 } x$, the range of values of $t$ is
$$\frac { \mathbf { E } } { \mathbf { F } } \leqq t \leqq \mathbf { G } .$$
When we express the right side of (2) in terms of $t$ and consider it as a function $g ( t )$, its derivative is
$$g ^ { \prime } ( t ) = \mathbf { H } t ^ { 2 } - \mathbf { I } t + \mathbf { J } .$$
Hence $f ( x )$ is maximized at $x = \mathbf { K L }$, and its maximum value is $\mathbf { M }$.

(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.

Write $( \sqrt [ 3 ] { 49 } ) ^ { 100 }$ in scientific notation as $( \sqrt [ 3 ] { 49 } ) ^ { 100 } = a \times 10 ^ { n }$, where $1 \leq a < 10$ and $n$ is a positive integer. If the integer part of $a$ is $m$, then the ordered pair $( m , n ) = ($ (25) )(26).

A sequence of five real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 } , a _ { 5 }$ where each term is greater than 1, and between any two adjacent terms, one is twice the other. If $a _ { 1 } = \log _ { 10 } 36$, how many possible values can $a _ { 5 }$ have?
(1) 3
(2) 4
(3) 5
(4) 7
(5) 8

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) $2 ^ { a } > 3 ^ { b } > \left( \frac { 1 } { 2 } \right) ^ { c }$

Let $a_1, a_2, a_3, a_4$ be a geometric sequence with first term 10 and common ratio 10. Let $b = \sum_{n=1}^{3} \log_{a_n} a_{n+1}$. Select the correct option.
(1) $2 < b \leq 3$
(2) $3 < b \leq 4$
(3) $4 < b \leq 5$
(4) $5 < b \leq 6$
(5) $6 < b \leq 7$

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) $3 a = 2 b$

On a coordinate plane, it is known that vector $\overrightarrow{PQ} = \left(\log \frac{1}{5}, -10^{-5}\right)$, where point $P$ has coordinates $\left(\log \frac{1}{2}, 2^{-5}\right)$. Select the correct option.
(1) Point $Q$ is in the first quadrant
(2) Point $Q$ is in the second quadrant
(3) Point $Q$ is in the third quadrant
(4) Point $Q$ is in the fourth quadrant
(5) Point $Q$ is on a coordinate axis

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) $c < b < 0$

Let $\Gamma$ be the graph formed by points $(x, y)$ satisfying $y = \log x$ on the coordinate plane. Which of the following relationships produce graphs that are completely identical to $\Gamma$?
(1) $y + \frac{1}{2} = \log(5x)$
(2) $2y = \log\left(x^{2}\right)$
(3) $3y = \log\left(x^{3}\right)$
(4) $x = 10^{y}$
(5) $x^{3} = 10^{\left(y^{3}\right)}$

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) $\frac { 1 } { \log b }$

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) $\frac { \log 11 } { \log 7 }$

$$\log_{3} 5 = a$$
Given this, what is the value of $\log_{5} 15$?
A) $\frac{a}{a+1}$
B) $\frac{a+1}{a}$
C) $\frac{a}{a+3}$
D) $\frac{a+3}{a}$
E) $\frac{4a}{3}$

$$\frac{1}{\log_{2} 6} + \frac{1}{\log_{3} 6}$$
Which of the following is this expression equal to?
A) $\frac{1}{3}$
B) $1$
C) $2$
D) $\log_{6} 2$
E) $\log_{6} 3$

$$0 \leq \log_{2}(x-5) \leq 2$$
How many integers $x$ satisfy these inequalities?
A) 2
B) 3
C) 4
D) 5
E) 6

For positive real numbers $a$, $b$, $c$ different from 1, $$\log_{a} b = \frac{1}{2}, \quad \log_{a} c = 3$$ Given this, what is the value of the expression $\log_{b}\left(\frac{b^{2}}{c\sqrt{a}}\right)$?
A) $\frac{3}{2}$
B) $\frac{5}{2}$
C) $\frac{5}{3}$
D) $-6$
E) $-5$

$12^{a} = 2$
$$6^{b} = 3$$
Given that, what is the value of the expression $\mathbf{12}^{\boldsymbol{(}\mathbf{1} - \mathbf{a}\mathbf{)2b}}$?
A) 15 B) 16 C) 9 D) 8 E) 4

$$\frac{2^{x^{2} - y^{2}}}{4^{x^{2} + xy}} = \frac{1}{2}$$
Given that, what is the value of the expression $(x + y)^{2}$?
A) 2 B) 4 C) 1 D) $\frac{1}{2}$ E) $\frac{1}{4}$

$$\log _ { 9 } \left( x ^ { 2 } + 2 x + 1 \right) = t \quad ( x > - 1 )$$
Given this equation, which of the following is the expression for x in terms of t?
A) $3 ^ { t } - 1$
B) $3 ^ { \mathrm { t } - 1 }$
C) $3 - 2 ^ { t }$
D) $2 \cdot 3 ^ { \mathrm { t } - 1 }$
E) $3 ^ { t } - 2$

Let $\mathbf { x }$ and $\mathbf { y }$ be real numbers.
$$2 ^ { x } - 2 ^ { -y } \left( 2 ^ { x+y } - 2 \right)$$
Which of the following is this expression equal to?
A) $2 ^ { x+1 }$
B) $2 ^ { y-x }$
C) $2 ^ { -y+1 }$
D) $\frac { 2 } { 9 }$
E) $\frac { 4 } { 9 }$