Evaluate $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\frac{1}{\sqrt{1368}+\sqrt{1369}}$.
How many integer pairs $(a,b)$ with $1 < a, b\le 2015$ are there such that $log_a b$ is an integer?
How many digits are there if the numbers $2^{2015}$ and $5^{2015}$ are written one after another?
What is the smallest positive number $x$ for which $\left(16^\sqrt{2}\right)^x$ represents a positive integer?
What are all values of $x$ for which $log_x\sqrt{x+12}>1$?
If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$
According to the standard convention for exponentiation, \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible?
For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Let $N=f(11)+f(13)+f(14)$. Which of the following relations is true?
For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$?
The graphs of $y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,$ and $y=\log_x \dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs?
What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$?
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer?
Let $x,y,$ and $z$ be real numbers satisfying the system $\log_2(xyz-3+\log_5 x)=5$ $\log_3(xyz-3+\log_5 y)=4$ $\log_4(xyz-3+\log_5 z)=4$ Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.
If $x$ is a number such that $3^x + 3^{x+2} = 9^x + 9^{x+2}$, then what is the value of $3^x$? Express your answer as a common fraction.
Compute the least possible non-zero value of $A^2+B^2+C^2$ such that $A, B,$ and $C$ are integers satisfying $A\log16+B\log18+C\log24=0$.
In $\triangle{LEO}$, point $J$ lies on $\overline{LO}$ such that $\overline{JE}\perp\overline{EO}$, and point $S$ lies on $\overline{LE}$ such that $\overline{JS}\perp\overline{LE}$. Given that $JS=9, EO=20,$ and $JO+SE=37$, compute the perimeter of $\triangle{LEO}$.
Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.
For each ordered pair of real numbers $(x,y)$ satisfying\[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that\[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$.
What is the value of \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\]
Positive real numbers $x\ne 1$ and $y\ne 1$ satisfy $\log_2x=\log_y16$ and $xy=64$. What is $\left(\log_2\frac{x}{y}\right)^2$?
Positive real numbers $a$ and $b$ have the property that $$\sqrt{\log a}+\sqrt{\log b} +\log\sqrt{a} + \log\sqrt{b}=100$$
and all four terms on the left are positive integers, where $\log$ denotes the base-$10$ logarithm. What is $ab$?
There is a unique positive integer $n$ such that $$\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}$$ What is the sum of the digits of $n?$
The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex?