Comparing Powers: A Mental Technique

[vin300]

This is by no means a trick question, I would want to know whether there is any technique to compare two powered terms with different bases, that is if I say I want to compare 2 powered 200 to 3 powered 120 to estimate which of these may be greater without any special or digital aid, how can I do it in my mind?

 

[TeethWhitener]

Take the logarithm of both numbers:

ln(2²⁰⁰) = 200 ln 2 = 138.629436112
ln(3¹²⁰) = 120 ln 3 = 131.83347464

So 2²⁰⁰ > 3¹²⁰

EDIT: fixed a math error.

 

[DuckAmuck]

Are you talking about exponents?
So like x^n vs y^m?
You could just take the logs, and do some algebra.

Have the ~ symbol be a placeholder for =,<,> symbols

$$2^{200} \sim 3^{120}$$
$$200 log(2) \sim 120 log(3)$$
$$\frac{200}{120} \sim \frac{log(3)}{log(2)}$$
$$\frac{5}{3} \sim log_2(3)$$
$$2^{\frac{5}{3}} \sim 3$$
$$2^5 \sim 3^3$$
$$32 \sim 27$$

therefore: ~ is >
so
$$2^{200} > 3^{120}$$

No calculator needed.

 

[vin300]

Thanks.