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# Logarithms and Exponents...

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I know the domain and range for $$f\left(x\right)=\log _6\left(\log _{\frac{1}{5}}\left(x\right)\right)$$. However, I don't completely understand how that was achieved. Can someone please help me?

Jan 24, 2022

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Well  for the first log1/5  portion,     x has to be > 0     there are no logs of negative numbers...or    log of  zero    so the   domain goes from    0 < x < +inf

the corresponding  RANGE can go from - inf   ( for very, very small x)  to + inf for very, very large x  values

the base of the logs is irrelavent

Then for the SECOND log portion   log6    again you cannot have negative numbers or zero    ....this means that 'x' has to be => 1 for the first log fxn (remember no negative numbers!)

then the range will change from log6(1) = 0   to  + infinity

so   in summary    domain      1<=x <= + inf

range        0 <= y <=+ inf

Jan 24, 2022