Logarithms: all problems

What is the value of $a$ for which $\frac{1}{\log_2 a} + \frac{1}{\log_3 a} + \frac{1}{\log_4 a} = 1$?

What is the value of

What is the value of $\left(625^{\log_5 2015}\right)^{\frac{1}{4}}$?

There is a unique positive integer $n$ such that

What is the sum of the digits of $n$?

There is a unique positive integer $n$ such that

What is the sum of the digits of $n$?

The sequence

is an arithmetic progression. What is $x$?

For how many positive integers $x$ is $\log_{10}(x - 40) + \log_{10}(60 - x) < 2$?

Positive real numbers $a$ and $b$ have the property that

and all four terms on the left are positive integers, where $\log$ denotes the base 10 logarithm. What is $ab$?

Which of the following is the value of $\sqrt{\log_2 6 + \log_3 6}$?

The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2 x = \log_y 16$ and $xy = 64$. What is $\left(\log_2 \frac{x}{y}\right)^2$?

What is the value of

What is the value of

What is the product of all the solutions to the equation

Let $S$ be the sum of all positive real numbers $x$ for which

Which of the following statements is true?

Suppose that $4^a = 5$, $5^b = 6$, $6^c = 7$, and $7^d = 8$. What is $a \cdot b \cdot c \cdot d$?

Real numbers $x$ and $y$ with $x, y > 1$ satisfy $\log_x\left(y^x\right) = \log_y\left(x^{4y}\right) = 10$. What is the value of $xy$?

The sum of the base-10 logarithms of the divisors of $10^n$ is $792$. What is $n$?

In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \ge 2$. A Martian student writes down

and finds that this system of equations has a single real number solution $x > 1$. Find $b$.

Let $S$ be the sum of the base 10 logarithms of all the proper divisors (all divisors of a number excluding itself) of $1000000$. What is the integer nearest to $S$?

Positive integers $a$ and $b$ satisfy the condition

Find the sum of all possible values of $a + b$.

The sequence $a_1, a_2, \dots$ is geometric with $a_1 = a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $\log_8 a_1 + \log_8 a_2 + \cdots + \log_8 a_{12} = 2006$, find the number of possible ordered pairs $(a, r)$.

If $\log\left(xy^3\right) = 1$ and $\log\left(x^2y\right) = 1$, what is $\log(xy)$?

The increasing geometric sequence $x_0, x_1, x_2, \dots$ consists entirely of integral powers of $3$. Given that

find $\log_3(x_{14})$.

The solution of the equation $7^{x+7} = 8^x$ can be expressed in the form $x = \log_b 7^7$. What is $b$?

For all integers $n$ greater than $1$, define $a_n = \frac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Then $b - c$ equals

The numbers $\log\left(a^3b^7\right)$, $\log\left(a^5b^{12}\right)$, and $\log\left(a^8b^{15}\right)$ are the first three terms of an arithmetic sequence, and the $12^{\text{th}}$ term of the sequence is $\log b^n$. What is $n$?

Determine the value of $ab$ if $\log_8 a + \log_4 b^2 = 5$ and $\log_8 b + \log_4 a^2 = 7$.

The number

can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The domain of the function $f(x) = \log_{\frac12}\left(\log_4\left(\log_{\frac14}\left(\log_{16}\left(\log_{\frac{1}{16}} x\right)\right)\right)\right)$ is an interval of length $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

The solutions to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\frac{1}{3}$ or $\frac{1}{2}$, can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$?

Let $m > 1$ and $n > 1$ be integers. Suppose that the product of the solutions for $x$ of the equation

is the smallest possible integer. What is $m + n$?

Let $S$ be the set of ordered triples $(x, y, z)$ of real numbers for which

There are real numbers $a$ and $b$ such that for all ordered triples $(x, y, z)$ in $S$ we have $x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}$. What is the value of $a + b$?