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 $a$ with $0 < a < 1$, when $10 ^ { a }$ is divided by 3, the quotient is an integer and the remainder is 2. What is the sum of all values of $a$? [4 points]
(1) $3 \log 2$
(2) $6 \log 2$
(3) $1 + 3 \log 2$
(4) $1 + 6 \log 2$
(5) $2 + 3 \log 2$

For a two-digit natural number $N$, when the mantissa of $\log N$ is $\alpha$,
$$\frac { 1 } { 2 } + \log N = \alpha + \log _ { 4 } \frac { N } { 8 }$$
Find the value of $N$ that satisfies this equation. [4 points]

For a natural number $n$, let $f ( n )$ be the mantissa of $\log n$. What is the number of elements in the set
$$A = \{ f ( n ) \mid 1 \leqq n \leqq 150 , n \text { is a natural number } \}$$
? [3 points]
(1) 131
(2) 133
(3) 135
(4) 137
(5) 139

For a natural number $n$ less than 10, when $\left( \frac { n } { 10 } \right) ^ { 10 }$ has a non-zero digit appearing for the first time in the sixth decimal place, what is the value of $n$? (Use $\log 2 = 0.3010 , \log 3 = 0.4771$ for calculations.) [4 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

For a natural number $A$, let the characteristic of $\log A$ be $n$ and the mantissa be $\alpha$. Find the number of values of $A$ such that $n \leqq 2\alpha$ holds. (Given: $3.1 < \sqrt { 10 } < 3.2$) [4 points]

For a positive number $x$, let the characteristic and mantissa of $\log x$ be $f ( x )$ and $g ( x )$, respectively. The number of natural numbers $n$ satisfying the two inequalities
$$f ( n ) \leq f ( 54 ) , \quad g ( n ) \leq g ( 54 )$$
is? [4 points]
(1) 42
(2) 44
(3) 46
(4) 48
(5) 50

For a positive real number $x$, let $f ( x )$ and $g ( x )$ be the characteristic and mantissa of $\log x$, respectively. For a natural number $n$, let $a _ { n }$ be the product of all values of $x$ satisfying $f ( x ) - ( n + 1 ) g ( x ) = n$. What is the value of $\lim _ { n \rightarrow \infty } \frac { \log a _ { n } } { n ^ { 2 } }$? [4 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

For a real number $x > 1$, let $f ( x )$ and $g ( x )$ be the characteristic and mantissa of $\log x$ respectively. When the value of $3 f ( x ) + 5 g ( x )$ is a multiple of 10, the values of $x$ are listed in increasing order. Let the 2nd value be $a$ and the 6th value be $b$. What is the value of $\log a b$? [4 points]
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16

For a positive number $x$, let $f ( x )$ be the characteristic (integer part) of $\log x$.
How many natural numbers $n$ not exceeding 100 satisfy $$f ( n + 10 ) = f ( n ) + 1$$ ? [4 points]
(1) 11
(2) 13
(3) 15
(4) 17
(5) 19

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 $\mathbf{L}$.

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).