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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.

Level 1

         

\[ \large \log_7 35 + \log_7 35 - \log_7 25 = \ ? \]

\[\large\log_{\color{green}{y}}{\color{blue}{x}}+\log_{\color{blue}{x}}{\color{green}{y}}=-2\] Find the value of \(\color{blue}{x}\color{green}{y}.\)

If \( 4 \log_{12} \left(x^9\right) + \log_{12} \left(x^6\right) = 126\), find \(x.\)

\[\frac{1}{\log_{2}{100!}}+\frac{1}{\log_{3}{100!}}+\frac{1}{\log_{4}{100!}}+\ldots+\frac{1}{\log_{100}{100!}}= \ ? \]

Find the smallest number among the following numbers:

(A) \(\log_{2015}2016\)

(B) \(\log_{2016}2017\)

(C) \(\log_{2017}2018\)

(D) \(\log_{2018}2019\)

(E) \(\log_{2019}2020\)

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