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Logarithmic inequalities

Source:

9/28/2010
Let nn be a positive integer, prove that :
(a) log⁡10(n+1)>310n+log⁡10n;\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;
(b) log⁡n!>3n10(12+13+⋯+1n−1). \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).
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