Comparing Powers of Different BasesHow can I know if one power is bigger than the...

Aaden Lam

Aaden Lam

Answered

2022-01-22

Comparing Powers of Different Bases
How can I know if one power is bigger than the other when the bases are different?
For example, considering 210 and 103 the former is the greater one, but how to prove this? Logarithms? I'll be working with big numbers, and though a more general solution is really appreciated, I will be comparing exactly powers of 2 and 10.

Answer & Explanation

Tyrn7i

Tyrn7i

Expert

2022-01-23Added 11 answers

log2 is the way to go.
log2(22000)=2000
log2(10800)=800log2(10)
So which is bigger 20 or 8log2(10)?
Let's see 20 is smaller than 8×3=24 and
log2(10)>log2(8)=log2(23)=3. So it looks like:
22000<10800
(no calculator required).
To compare:
3=log2(23)=log2(8)<log2(10)<log2(16)=log2(24)=4

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