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.

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 \( \log_z w \).

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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