Given that
$$2^{3x} = 8^{(y+3)}$$
and
$$4^{(x+1)} = \frac{16^{(y+1)}}{8^{(y+3)}}$$
what is the value of $x + y$?
A $-23$
B $-22$
C $-15$
D $-14$
E $-11$
F $-10$

A student wishes to evaluate the function $\mathrm { f } ( x ) = x \sin x$, where $x$ is in radians, but has a calculator that only works in degrees.
What could the student type into their calculator to correctly evaluate $\mathrm { f } ( 4 )$ ?
A $( \pi \times 4 \div 180 ) \times \sin ( 4 )$
B $( \pi \times 4 \div 180 ) \times \sin ( \pi \times 4 \div 180 )$
C $4 \times \sin ( \pi \times 4 \div 180 )$
D $( 180 \times 4 \div \pi ) \times \sin ( 4 )$
E $\quad ( 180 \times 4 \div \pi ) \times \sin ( 180 \times 4 \div \pi )$ F $\quad 4 \times \sin ( 180 \times 4 \div \pi )$

Find the positive difference between the two real values of $x$ for which
$$(\log_2 x)^4 + 12(\log_2(\frac{1}{x}))^2 - 2^6 = 0$$
A $4$
B $16$
C $\frac{15}{4}$
D $\frac{17}{4}$
E $\frac{255}{16}$
F $\frac{257}{16}$

Consider the following functions defined for $x > 1$ :
$$\begin{aligned} & \mathrm { f } ( x ) = \log _ { 2 } \left( \log _ { 2 } \sqrt { x } \right) \\ & \mathrm { g } ( x ) = \log _ { 2 } \left( \sqrt { \log _ { 2 } x } \right) \end{aligned}$$
Which one of the following is true for all values of $x > 1$ ?
A $0 \leq \mathrm { f } ( x ) \leq \mathrm { g } ( x )$ or $\mathrm { g } ( x ) \leq \mathrm { f } ( x ) \leq 0$
B $0 \leq \mathrm { g } ( x ) \leq \mathrm { f } ( x )$ or $\mathrm { f } ( x ) \leq \mathrm { g } ( x ) \leq 0$
C $\frac { 1 } { 2 } \leq \mathrm { f } ( x ) \leq \mathrm { g } ( x )$ or $\mathrm { g } ( x ) \leq \mathrm { f } ( x ) \leq \frac { 1 } { 2 }$
D $\frac { 1 } { 2 } \leq \mathrm { g } ( x ) \leq \mathrm { f } ( x )$ or $\mathrm { f } ( x ) \leq \mathrm { g } ( x ) \leq \frac { 1 } { 2 }$
E $1 \leq \mathrm { f } ( x ) \leq \mathrm { g } ( x )$ or $\mathrm { g } ( x ) \leq \mathrm { f } ( x ) \leq 1$ F $\quad 1 \leq \mathrm { g } ( x ) \leq \mathrm { f } ( x )$ or $\mathrm { f } ( x ) \leq \mathrm { g } ( x ) \leq 1$

Given that
$$\int _ { \log _ { 2 } 5 } ^ { \log _ { 2 } 20 } x \mathrm {~d} x = \log _ { 2 } M$$
what is the value of $M$ ?

The real numbers $x , y$ and $z$ are all greater than 1 , and satisfy the equations
$$\log _ { x } y = z \quad \text { and } \quad \log _ { y } z = x$$
Which one of the following equations for $\log _ { z } x$ must be true?
A $\quad \log _ { z } x = y$
B $\quad \log _ { z } x = \frac { 1 } { y }$
C $\log _ { z } x = x y$
D $\log _ { z } x = \frac { 1 } { x y }$
E $\quad \log _ { z } x = x z$ F $\log _ { z } x = \frac { 1 } { x z }$ G $\log _ { z } x = y z$ H $\log _ { z } x = \frac { 1 } { y z }$

Consider the following equation where $a$ is a real number and $a > 1$ :
$$( * ) \quad a ^ { x } = x$$
Which of the following equations must have the same number of real solutions as $( * )$ ? I $\quad \log _ { a } x = x$ II $\quad a ^ { 2 x } = x ^ { 2 }$ III $a ^ { 2 x } = 2 x$
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

Answer the following questions. You may use the fact that $0.3 < \log_{10} 2 < 0.31$ if necessary.

• [(1)] Find the smallest natural number $n$ such that $5^n > 10^{19}$.
  
• [(2)] Find the smallest natural number $m$ such that $5^m + 4^m > 10^{19}$.

$$\log_{3} 5 = a$$
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) $\log_{6} 3$

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) $-5$

$$\log _ { 9 } \left( x ^ { 2 } + 2 x + 1 \right) = t \quad ( x > - 1 )$$
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) $3 ^ { t } - 2$

$$\frac { 3 ^ { x } } { 2 ^ { 2 x } } = \frac { 1 } { 5 }$$
Given this, what is the value of the expression $5 ^ { \frac { 1 } { x } }$?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 9 } { 4 }$
D) $\frac { 9 } { 5 }$
E) $\frac { 5 } { 6 }$

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) $\frac { 4 } { 9 }$

$$\log _ { 2 } 3x + \log _ { 4 } x ^ { 2 } = 2$$
What is the value of x that satisfies the equation?
A) $\frac { \sqrt { 2 } } { 2 }$
B) $\frac { 3 \sqrt { 2 } } { 2 }$
C) $\frac { 5 \sqrt { 2 } } { 2 }$
D) $\frac { \sqrt { 3 } } { 3 }$
E) $\frac { 2 \sqrt { 3 } } { 3 }$

$$\begin{aligned} & 2 ^ { x } = \frac { 1 } { 5 } \\ & 3 ^ { y } = \frac { 1 } { 4 } \end{aligned}$$
Given this, what is the value of the product $x \cdot y$?
A) $\frac { \ln 3 } { \ln 2 }$
B) $\frac { \ln 15 } { \ln 2 }$
C) $\frac { \ln 5 } { \ln 4 }$
D) $\frac { \ln 25 } { \ln 3 }$
E) $\frac { \ln 5 } { \ln 6 }$

For real numbers x and y
$$2 ^ { x } = 6 ^ { x + y - 1 }$$
Given this, what is $3 ^ { \mathbf { x } }$ in terms of y?
A) $3 ^ { 1 - y }$
B) $6 ^ { 1 - y }$
C) $6 ^ { y }$
D) $9 ^ { - y }$
E) $9 ^ { 1 + y }$

$$\log _ { 8 } \left( \log _ { 9 } ( \sqrt { x + 1 } ) \right) = \frac { - 2 } { 3 }$$
Given this, what is x?
A) 2
B) 3
C) 5
D) 7
E) 8

$$\begin{aligned} & f ( x ) = - \log _ { 2 } x \\ & g ( x ) = \log _ { 10 } x \end{aligned}$$
Given this, what is the value of a that satisfies the equality $\left( \right.$ gof $\left. ^ { - 1 } \right) ( a ) = \ln 2$?
A) $\ln 2$
B) $\frac { \ln 2 } { \ln 10 }$
C) $\frac { \ln 10 } { \ln 2 }$
D) $\ln \left( \frac { 1 } { 10 } \right)$
E) $\ln \left( \frac { 1 } { 2 } \right)$

For the function $f ( x ) = \log _ { x } 2$,
$$f \left( 4 ^ { a } \right) \cdot f ^ { - 1 } \left( \frac { 1 } { 3 } \right) = 6$$
What is the value of a that satisfies this equation?
A) $\frac { 1 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 4 } { 3 }$

$$\log _ { 2 } \left( \frac { 1 } { \sqrt { x } } \right) + \log _ { 4 } \left( \frac { 4 } { y } \right) = 3$$
Given that, what is the product $x \cdot y$?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 3 } { 8 }$
E) $\frac { 1 } { 16 }$

$$\lim _ { x \rightarrow \infty } \frac { \ln ( x - 3 ) } { \ln \sqrt { x } }$$
What is the value of this limit?
A) 1
B) 2
C) 3
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 2 }$

$\log _ { 4 } x$ and $\log _ { 4 } \left( x ^ { 2 } \right)$ are consecutive two positive even integers.
Accordingly, what is the value of $\log _ { x } 4$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 16 }$
D) 1
E) 2

Let k be a positive real number such that for the function
$$f ( x ) = \log _ { x } ( x - k )$$
we have $f ( 3 k ) = \frac { 2 } { 3 }$. What is k?
A) $\frac { 3 } { 8 }$
B) $\frac { 9 } { 8 }$
C) $\frac { 27 } { 8 }$
D) $\frac { 2 } { 9 }$
E) $\frac { 4 } { 9 }$

Let t be a real number. The equalities
$$\begin{aligned} & x = e ^ { 2 \cos t } \\ & y = e ^ { 3 \sin t } \end{aligned}$$
are given.
Accordingly, which of the following gives the relationship between $x$ and y that is satisfied for every real number t?
A) $\ln ^ { 2 } x + \ln ^ { 2 } y = 4$
B) $\ln ^ { 2 } x + \ln ^ { 2 } y = 9$
C) $9 \ln ^ { 2 } x + 2 \ln ^ { 2 } y = 27$
D) $\ln ^ { 2 } x + 4 \ln ^ { 2 } y = 28$
E) $9 \ln ^ { 2 } x + 4 \ln ^ { 2 } y = 36$