Log problem: Which one is larger, 2^500 or 5^200?

[dodoje]

Solving a log problem

Which one is larger, 2^500 or 5^200?

Is the solution method for this something related to (bigger no.) - (smaller no.) = (positive no.) ?

Thank you!

 

[arthur ohlsten]

2^500>?5^200

let y=500log2
let z=200log 5

is y>z?

500 log 2 >? 200log 5

divide by 200

2.5 log 2>? log 5
log 2^2.5 >? log5
log 2^(5/2) >?log 5

raise to power of 10

square root [2^5] >? 5
sqrt32>?5
5.7 >? 5 yes answer

please check for errors

Arthur

 

[soroban]

Re: Log problem

Hello, dodoje!

Don't need logs for this one . . .

Which is larger: $\displaystyle \,2^{500}$ or $\displaystyle 5^{200}$ ?

We have: $\displaystyle \:2^{500}\;<?>\;5^{200}$


Take the $\displaystyle 100^{th}$ root of both sides:

. . $\displaystyle \left(2^{500}\right)^{\frac{1}{100}}\;<?>\;\left(5^{200}\right)^{\frac{1}{100}}\;\;\Rightarrow\;\;2^5\;<?>\;5^2$


Since $\displaystyle 32\:>\:25$, the inequality is "greater than".


Therefore: $\displaystyle \:2^{500}\;> \;5^{200}$