If $\arcsin x = \ln y$, then $\frac { d y } { d x } =$
(A) $\frac { y } { \sqrt { 1 - x ^ { 2 } } }$
(B) $\frac { x y } { \sqrt { 1 - x ^ { 2 } } }$
(C) $\frac { y } { 1 + x ^ { 2 } }$
(D) $e ^ { \arcsin x }$
(E) $\frac { e ^ { \arcsin x } } { 1 + x ^ { 2 } }$

A equação $\log_2(x+1) = 3$ tem como solução
(A) $x = 5$ (B) $x = 6$ (C) $x = 7$ (D) $x = 8$ (E) $x = 9$

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

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 logarithmic equation $\left( \log _ { 2 } x \right) ^ { 2 } - 4 \log _ { 2 } x = 0$, let the two roots be $\alpha , \beta$ respectively. Find the value of $\alpha + \beta$. [3 points]

For two real numbers $a , b$ with $1 < a < b$,
$$\frac { 3 a } { \log _ { a } b } = \frac { b } { 2 \log _ { b } a } = \frac { 3 a + b } { 3 }$$
holds. Find the value of $10 \log _ { a } b$. [3 points]

When $\alpha$ is the root of the logarithmic equation $\log _ { 3 } ( x - 4 ) = \log _ { 9 } ( 5 x + 4 )$, find the value of $\alpha$. [3 points]

Find the value of $x$ that satisfies the equation $\log _ { 3 } ( x - 11 ) = 3 \log _ { 3 } 2$. [3 points]

Solve the logarithmic equation $\log _ { 2 } ( x + 6 ) = 5$. [3 points]

For two real numbers $a , b$ greater than 1, $$\log _ { \sqrt { 3 } } a = \log _ { 9 } a b$$ holds. Find the value of $\log _ { a } b$. [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For natural numbers $n \geq 2$, what is the sum of all values of $n$ such that $5 \log _ { n } 2$ is a natural number? [4 points]
(1) 34
(2) 38
(3) 42
(4) 46
(5) 50

Solve the equation $$\log _ { 2 } ( 3 x + 2 ) = 2 + \log _ { 2 } ( x - 2 )$$ for the real number $x$. [3 points]

Solve the equation $$\log_{2}(x - 3) = \log_{4}(3x - 5)$$ for the real number $x$. [3 points]

14. Given that $f ( x )$ is an odd function, and when $x < 0$, $f ( x ) = - \mathrm { e } ^ { a x }$. If $f ( \ln 2 ) = 8$, then $a =$ $\_\_\_\_$ .

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$ then $x$ equals
(A) $4 ^ { 10 }$
(B) 100
(C) $\log _ { 10 } 4$
(D) none of the above

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$, then $x$ equals
(a) $4 ^ { 10 }$
(b) 100
(c) $\log _ { 10 } 4$
(d) none of the above.

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$, then $x$ equals
(a) $4 ^ { 10 }$
(b) 100
(c) $\log _ { 10 } 4$
(d) none of the above.

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$ then $x$ equals
(A) $4 ^ { 10 }$
(B) 100
(C) $\log _ { 10 } 4$
(D) none of the above

If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$ then $x$ equals
(A) $4 ^ { 10 }$
(B) 100
(C) $\log _ { 10 } 4$
(D) none of the above

Let $a, b$ and $c$ be real numbers, each greater than 1, such that $$\frac{2}{3}\log_b a + \frac{3}{5}\log_c b + \frac{5}{2}\log_a c = 3$$ If the value of $b$ is 9, then the value of $a$ must be
(A) $\sqrt[3]{81}$
(B) $\frac{27}{2}$
(C) 18
(D) 27.

47. Let $\left( x _ { 0 } , y _ { 0 } \right)$ be the solution of the following equations
$$\begin{aligned} ( 2 x ) ^ { \ln 2 } & = ( 3 y ) ^ { \ln 3 } \\ 3 ^ { \ln x } & = 2 ^ { \ln y } . \end{aligned}$$
Then $x _ { 0 }$ is
(A) $\frac { 1 } { 6 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 1 } { 2 }$
(D) 6

ANSWER: C

1. The value of $\int _ { \sqrt { \ln 2 } } ^ { \sqrt { \ln 3 } } \frac { x \sin x ^ { 2 } } { \sin x ^ { 2 } + \sin \left( \ln 6 - x ^ { 2 } \right) } d x$ is
   (A) $\frac { 1 } { 4 } \ln \frac { 3 } { 2 }$
   (B) $\frac { 1 } { 2 } \ln \frac { 3 } { 2 }$
   (C) $\ln \frac { 3 } { 2 }$
   (D) $\frac { 1 } { 6 } \ln \frac { 3 } { 2 }$

ANSWER: A

Let $a = 3 \sqrt { 2 }$ and $b = \frac { 1 } { 5 ^ { 1 / 6 } \sqrt { 6 } }$. If $x , y \in \mathbb { R }$ are such that
$$\begin{aligned} & 3 x + 2 y = \log _ { a } ( 18 ) ^ { \frac { 5 } { 4 } } \\ & 2 x - y = \log _ { b } ( \sqrt { 1080 } ) \end{aligned}$$
then $4 x + 5 y$ is equal to $\_\_\_\_$ .

If the sum of the first 20 terms of the series $\log_{(7^{1/2})}x + \log_{(7^{1/3})}x + \log_{(7^{1/4})}x + \ldots$ is 460, then $x$ is equal to:
(1) $7^2$
(2) $7^{1/2}$
(3) $e^2$
(4) $7^{46/21}$

The inverse of $y = 5 ^ { \log x }$ is:
(1) $x = 5 ^ { \log y }$
(2) $x = y ^ { \log 5 }$
(3) $y = x ^ { \frac { 1 } { \log 5 } }$
(4) $x = 5 ^ { \frac { 1 } { \log y } }$

The number of solutions of the equation $\log _ { 4 } ( x - 1 ) = \log _ { 2 } ( x - 3 )$ is