Logarithmic Functions

Logarithmic Functions: Level 4 Challenges


log(tan1)+log(tan2)+log(tan3)++log(tan89)= ? \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

(log3x)(log4x)(log5x)=(log3x)(log4x)+(log4x)(log5x)+(log3x)(log5x). (\log_3x)(\log_4x)(\log_5x) = (\log_3x)(\log_4x) +(\log_4x)(\log_5x) +(\log_3x)(\log_5x).

It is given that log6a+log6b+log6c=6\log_{6}a + \log_{6}b + \log_{6}c = 6, where aa, bb and cc are positive integers that form an increasing geometric sequence and bab - a is the square of an integer.

Find a+b+c.a + b + c.

The value of x\displaystyle{x} that satisfies the equation log2(2x1+3x+1)=2xlog2(3x)\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x) can be expressed as logab\displaystyle{\log_{a}b}. What is the value of a+b\displaystyle{a+b}?

\quad Note: aa and bb are both fractions and their sum is an integer.

log2(log2x(log2y(21000)))=0\large \log_2(\log_{2^x}(\log_{2^y}(2^{1000}))) = 0

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


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