<s>[INST] Solving $5^n > 4,000,000$ without a calculator 
If $n$ is an integer and $5^n > 4,000,000.$ What is the least possible value of $n$? (answer: $10$)

How could I find the value of $n$ without using a calculator ?
 [/INST] By logarithm rules: 
$$5^{n}>4\cdot10^{6}\iff n>\log_{5}2^{2}2^{6}5^{6}=\log_{5}2^{8}+\log_{5}5^{6}=\log_{5}2^{8}+6=\log_{5}256+6$$
Since these are relatively small numbers I assume it is ok to write
: $5^{3}=125$ thus clearly $3<\log_{5}256<4$ hence the minimal $n$
that satisfies this inequality is $4+6=10$ 
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