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`.