8. Let $a$ and $b$ be positive numbers not equal to $1$. Then ``$3 ^ { a } > 3 ^ { b } > 3$'' is ``$\log _ { a } 3 < \log _ { b } 3$'' a
(A) necessary and sufficient condition
(B) sufficient but not necessary condition
(C) necessary but not sufficient condition
(D) neither sufficient nor necessary condition

Let $a = \log _ { 0.2 } 0.3, b = \log _ { 2 } 0.3$, then
A. $a + b < ab < 0$
B. $ab < a + b < 0$
C. $a + b < 0 < ab$
D. $ab < 0 < a + b$

Let $f ( x )$ be an even function with domain $\mathbf { R }$ that is monotonically decreasing on $( 0 , + \infty )$. Then
A. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right)$
B. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right)$
C. $f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$
D. $f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$

Let $a = \log _ { 3 } 2 , b = \log _ { 5 } 3 , c = \frac { 2 } { 3 }$. Then
A. $a < c < b$
B. $a < b < c$
C. $b < c < a$
D. $c < a < b$

Given $5 ^ { 5 } < 8 ^ { 4 } , 13 ^ { 4 } < 8 ^ { 5 }$ . Let $a = \log _ { 5 } 3 , b = \log _ { 8 } 5 , c = \log _ { 13 } 8$ , then
A. $a < b < c$
B. $b < a < c$
C. $b < c < a$
D. $c < a < b$

7. Given $a = \log _ { 5 } 2 , b = \log _ { 8 } 3 , c = \frac { 1 } { 2 }$, which of the following judgments is correct? ( )
A. $c < b < a$
B. $b < a < c$
C. $a < c < b$
D. $a < b < c$
【Answer】C 【Solution】 【Analysis】Use the monotonicity of logarithmic functions to compare the sizes of $a$, $b$, and $c$, and thus reach the conclusion. 【Detailed Solution】 $a = \log _ { 5 } 2 < \log _ { 5 } \sqrt { 5 } = \frac { 1 } { 2 } = \log _ { 8 } 2 \sqrt { 2 } < \log _ { 8 } 3 = b$, that is, $a < c < b$. Therefore, the answer is: C.

Given $a = \log _ { 5 } 2 , b = \log _ { 8 } 3 , c = \frac { 1 } { 2 }$, which of the following judgments is correct?
A. $c < b < a$
B. $b < a < c$
C. $a < c < b$
D. $a < b < c$

7. Let $a = 0.1 \mathrm { e } ^ { 0.1 }$, $b = \frac { 1 } { 9 }$, $c = - \ln 0.9$. Then
A. $a < b < c$
B. $c < b < a$
C. $c < a < b$
D. $a < c < b$

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

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