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If $x, \log_{10}(x) , \log_{10}\log_{10}(x)$ are in arithmetic progression, find the range of $x.$

(a) $0 < x < 1$

(b) $1 < x < 10$

(c) $10 < x < 100$

(d) $100 < x < 1000$

I have found the answer but I want a solution using logic without graph.

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    I re-formatted your question, please double check I didn't change anything.2012-08-19
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    What is the base of the log ?2012-08-19
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    @Belgi $\log_{10} $2012-08-19
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    In case anyone is wondering, the progression turns out to be approximately $1.22802, 0.0892054, -1.04961$.2012-08-19

4 Answers 4