Comparing Powers of Different BasesHow 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.

Question

Comparing Powers of Different BasesHow 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&#039;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.

Tyrn7i · Accepted Answer

log2 is the way to go.log2(22000)=2000log2(10800)=800log2(10)So which is bigger 20 or 8log2(10)?Let&#039;s see 20 is smaller than 8×3=24 andlog2(10)&amp;gt;log2(8)=log2(23)=3. So it looks like:22000&amp;lt;10800(no calculator required).To compare:3=log2(23)=log2(8)&amp;lt;log2(10)&amp;lt;log2(16)=log2(24)=4