Question

use the fact that 10^3 = 2^10 to mentally find a vaule of n for which 2^n > 10^50 & 10^n > 2^300

Answer 1

$2^n>10^{50}=10^{3\cdot16+2}=10^2(10^3)^{16}=10^2(2^{10})^{16}=5^22^{162}$
$2^{n-162}>25$

The least power of 2 that exceeds 25 is $2^5=32$, so we have

$2^{n-162}=2^N>25\implies N=5\implies n-162=5\implies n=167$

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$10^n>2^{300}=(2^{10})^{30}=(10^3)^{30}=10^{90}$

The least integer $n$ that satisfies this inequality would clearly be $n=91$.