Logarithmic Functions
Let \(w, x, y, z \) be positive reals greater than 1, and
\[ \log_x w = 24, \log_y w = 40, \log_{xyz} w = 12 .\]
Evaluate \( \log_z w \).
The only real solution of the equation \[\ln x + \ln (x^2-34)=\ln 72\] can be uniquely written in the form \(a+\sqrt{b}\) for some integers \(a\) and \(b\). Find \(a+b\).
Given that \(\log_{10}2=0.3010\), \(\log_{10}3=0.4771\) and \(\log_{10}7=0.8451\), let \(a\) be the number of digits in \(3^{62}\). If \(b\) is the first digit of \(3^{62}\), what is \(a+b\)?
Let \(N\) be the number of times that the first digit of \(2^n\) is \(1,\) with \(n \) being an integer such that \(0\leq n \leq 10^{10}\). What are the last three digits of \(N?\)
You may use the fact that
\[ 0.3010299956639811952 < \log_{10} 2 < 0.3010299956639811953 \]
Evaluate \[\lfloor \log_{8} 1 \rfloor + \lfloor \log_{8} 2 \rfloor + \lfloor \log_{8} 3 \rfloor + \cdots + \lfloor \log_{8} 154 \rfloor.\]
Details and assumptions
Greatest Integer Function / Floor Function: The function \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer smaller than or equal to \(x\). For example, \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).