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.

\[\large \log_{\sqrt{x}} \left( \sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}} \right) = \, ? \]

- Clarification: \(x\) is a positive real number and \(x \neq 1.\)

How many digits does the number \( 2^{1000} \) contain?

You are given that \(\log_{10} 2 = 0.3010 \) correct up to 4 decimal places.

\[ \Large x ^{\log_{10} x } = 100x \]

How many real solutions are there to the above equation?

\[\Large\begin{cases} \log_x w & = & 24 \\ \log_y w & = & 40 \\ \log_{xyz} w & = & 12 \end{cases}\]

If \(x,y,z\) are real numbers greater than 1 and \(w\) is a positive number satisfying the system above, then find the value of \(\left (\log_w z \right)^{-1}\).

Given that

\[\log_2(\log_8x)=\log_8(\log_2x),\]

find the value of \((\log_2x)^2\).

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