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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.

# Logarithmic Functions: Level 4 Challenges

$\log(\tan 1^\circ ) + \log (\tan 2^\circ) + \log (\tan 3^\circ) + \ldots + \log (\tan 89^\circ ) = \ ?$

Find the sum of the roots of the equation

$(\log_3x)(\log_4x)(\log_5x) = (\log_3x)(\log_4x) +(\log_4x)(\log_5x) +(\log_3x)(\log_5x).$

It is given that $$\log_{6}a + \log_{6}b + \log_{6}c = 6$$, where $$a$$, $$b$$ and $$c$$ are positive integers that form an increasing geometric sequence and $$b - a$$ is the square of an integer.

Find $$a + b + c.$$

The value of $$\displaystyle{x}$$ that satisfies the equation $\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x)$ can be expressed as $$\displaystyle{\log_{a}b}$$. What is the value of $$\displaystyle{a+b}$$?

$$\quad$$ Note: $$a$$ and $$b$$ are both fractions and their sum is an integer.

$\large \log_2(\log_{2^x}(\log_{2^y}(2^{1000}))) = 0$

If $$x$$ and $$y$$ are positive integers satisfying the equation above, then find the sum of all possible values of $$x+y$$.

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