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Logarithmic Functions

Logarithmic scales are used when the range of possible values is very wide, such as the intensity of an earthquake or the acidity of a liquid, in order to avoid the use of large numbers.

Problem Solving

         

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 \(\lfloor -5 \rfloor = -5\).

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