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algebra regarding logs

log_3 (log 3x)=1. I have another 72 questions similar to this. can you answer this question?

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Nataliya D. | Patient and effective tutor for your most difficult subject.Patient and effective tutor for your mos...

Chris, you have to clarify the base of second "inner" logarithm. I might assume that bases are the same, and the problem will be:

log3 (log3 3x) = 1 , and according to properties of logarithm [logb b = 1 and log(ab) = log (a) + log (b)]
we have:

log3 3  +  log3 x  = 3

1 + log3 x  =  3 

log3 x = 2 

32 = x 

x = 9 


If no base is specified then it is understood to be base 10

Yes, it is. But how about human factor?

I just was watching international standards (ISO 31), that defines mathematical signs and symbols for use in physical sciences and technology. Sign for logarithm with base 10 is "lg" , and many countries are using this sign. For me it makes more sense, because it's more likely to forget write the base, than forget to write "o" :)
Anyway, our recipient has 2 versions of the answer ...

Interesting. I never seen that notation in any of my text books from college or high school for that matter but I have seen it on another question around here and I thought that was a typo. I learn something new everyday, thank you for sharing that I'll be sure to remember :). Yes he does, hopefully they are both helpful.

You're very welcome, Xavier!
Me too :) is learning something new here ...

Xavier J. | Tutor in Math, topics range from Algebra to Calculus.Tutor in Math, topics range from Algebra...
5.0 5.0 (3 lesson ratings) (3)

In order to answer this problem you need to be able to convert from log form to exponential form. In general if you have loga(b) = c then the exponential form of that would be ac = b.

So in your problem you have log3(log(3x)) = 1. Here a = 3, b = log(3x) and c = 1 so to convert this to exponential you get 31 = log(3x).

We can repeat this process again to get rid of the other log, this time with a = 10 b = 3x and c = 3. Doing so gives you 3x = 103 and x = 103/3