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

If for $x \in \left( 0 , \frac { \pi } { 2 } \right) , \log _ { 10 } \sin x + \log _ { 10 } \cos x = - 1$ and $\log _ { 10 } ( \sin x + \cos x ) = \frac { 1 } { 2 } \left( \log _ { 10 } n - 1 \right) , n > 0$, then the value of $n$ is equal to :
(1) 20
(2) 12
(3) 9
(4) 16

If sum of the first 21 terms of the series $\log _ { 9^{1/2} } x + \log _ { 9^{1/3} } x + \log _ { 9^{1/4} } x + \ldots$ where $x > 0$ is 504, then $x$ is equal to
(1) 243
(2) 9
(3) 7
(4) 81

The number of solutions of the equation $\log _ { ( x + 1 ) } \left( 2 x ^ { 2 } + 7 x + 5 \right) + \log _ { ( 2 x + 5 ) } ( x + 1 ) ^ { 2 } - 4 = 0 , x > 0$, is

For three positive integers $p , q , r , x ^ { p q ^ { 2 } } = y ^ { q r } = z ^ { p ^ { 2 } r }$ and $r = p q + 1$ such that 3, $3 \log _ { y } x , 3 \log _ { z } y , 7 \log _ { x } z$ are in A.P. with common difference $\frac { 1 } { 2 }$. The $r - p - q$ is equal to
(1) 2
(2) 6
(3) 12
(4) - 6

Let $a, b, c$ be three distinct positive real numbers such that $2a^{\log_e a} = bc^{\log_e b}$ and $b^{\log_e 2} = a^{\log_e c}$. Then $6a + 5bc$ is equal to $\_\_\_\_$.

The sum of all the solutions of the equation $( 8 ) ^ { 2 x } - 16 \cdot ( 8 ) ^ { x } + 48 = 0$ is :
(1) $1 + \log _ { 8 } ( 6 )$
(2) $1 + \log _ { 6 } ( 8 )$
(3) $\log _ { 8 } ( 6 )$
(4) $\log _ { 8 } ( 4 )$

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

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) $3 a = 2 b$

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)

5. Given that $y = - \log _ { 10 } ( 1 - x )$ for $x < 1$, find $x$ in terms of $y$.
A $\quad x = - \frac { 1 } { \log _ { 10 } ( 1 - y ) }$
B $x = 1 + \log _ { 10 } y$
C $x = 1 - \log _ { 10 } y$
D $\quad x = 1 - 10 ^ { - y }$
E $\quad x = 10 ^ { - y } - 1$
F $\quad x = 10 ^ { 1 - y }$

8. Given that $a ^ { x } b ^ { 2 x } c ^ { 3 x } = 2$, where $a , b$, and $c$ are positive real numbers, then $x =$
A $\quad \log _ { 10 } \left( \frac { 2 } { a + 2 b + 3 c } \right)$
B $\frac { \log _ { 10 } 2 } { \log _ { 10 } ( a + 2 b + 3 c ) }$
C $\quad \frac { 2 } { \log _ { 10 } ( a + 2 b + 3 c ) }$
D $\frac { 2 } { a + 2 b + 3 c }$
E $\quad \log _ { 10 } \left( \frac { 2 } { a b ^ { 2 } c ^ { 3 } } \right)$ F $\quad \frac { \log _ { 10 } 2 } { \log _ { 10 } \left( a b ^ { 2 } c ^ { 3 } \right) }$ G $\quad \frac { 2 } { \log _ { 10 } \left( a b ^ { 2 } c ^ { 3 } \right) }$ H $\frac { 2 } { a b ^ { 2 } c ^ { 3 } }$

The real roots of the equation $4 ^ { 2 x } + 12 = 2 ^ { 2 x + 3 }$ are $p$ and $q$, where $p > q$. The value of $p - q$ can be expressed as
A $\frac { 3 } { 4 }$ B 1 C 4 D $- \frac { 1 } { 2 } + \log _ { 10 } \frac { 3 } { 2 }$ E $\frac { \log _ { 10 } 3 } { \log _ { 10 } 4 }$ F $\frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$

Given the simultaneous equations
$$\begin{gathered} \log _ { 10 } 2 + \log _ { 10 } ( y - 1 ) = 2 \log _ { 10 } x \\ \log _ { 10 } ( y + 3 - 3 x ) = 0 \end{gathered}$$
the values of $y$ are
A $\frac { 5 } { 2 } \pm \frac { 3 \sqrt { 5 } } { 2 }$ B $3 \pm \sqrt { 3 }$ C $7 \pm 3 \sqrt { 3 }$ D 3,9 E 1,13

The line $y = m x + 4$ passes through the points ( $3 , \log _ { 2 } p$ ) and ( $\log _ { 2 } p , 4$ ). What are the possible values of $p$ ?
A $p = 1$ and $p = 4$
B $p = 1$ and $p = 16$
C $\quad p = \frac { 1 } { 4 } \quad$ and $\quad p = 4$
D $\quad p = \frac { 1 } { 4 } \quad$ and $\quad p = 64$
E $\quad p = \frac { 1 } { 64 }$ and $p = 4$
F $\quad p = \frac { 1 } { 64 }$ and $p = 16$

Find the sum of the real values of $x$ that satisfy the simultaneous equations:
$$\begin{aligned} \log_3(xy^2) &= 1 \\ (\log_3 x)(\log_3 y) &= -3 \end{aligned}$$

Find the real non-zero solution to the equation
$$\frac{2^{(9^x)}}{8^{(3^x)}} = \frac{1}{4}$$

The numbers $a , b$ and $c$ are each greater than 1 .
The following logarithms are all to the same base:
$$\begin{aligned} \log \left( a b ^ { 2 } c \right) & = 7 \\ \log \left( a ^ { 2 } b c ^ { 2 } \right) & = 11 \\ \log \left( a ^ { 2 } b ^ { 2 } c ^ { 3 } \right) & = 15 \end{aligned}$$
What is this base?

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 $-10$

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 $\frac{257}{16}$

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 $5^m + 4^m > 10^{19}$.

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

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

$$\log _ { 2 } 3x + \log _ { 4 } x ^ { 2 } = 2$$
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) $\frac { 2 \sqrt { 3 } } { 3 }$