What is the easiest way to compare two powers with different bases, signs, and exponants?
An example would help clarify this question.
How I'm reading it, you want to know a quick way to tell which quantity is larger when given something like `12^10` vs. `10^12`
Let's look at some smaller examples.
Maybe the larger base always wins?
I guess not.
Exponents are crazy powerful. An exponential function will always pass up a linear function, though sometimes it's hard to guess when. Look at this: y = 10x vs. y = 1.1^x. The red graph is very steep, and it's hard to believe that the orange graph will ever catch up and pass it.
But if we zoom far out:
Now I know you're not asking about linear quantities, but this just shows how powerful the exponent is.
Let's look back at `10^12` vs `12^10`
You know they're each larger than `10^10` , but can we get an idea of how much larger?
`10^12=10^10*10^2` so this one's 100 times larger.
`12^10=(10*1.2)^10=10^10*1.2^10` Since 1.2 is pretty close to 1, raising it to the 10th power won't make it that big.
So 10^12 wins.
There aren't any good rules that I know of for comparing exponential quantities, but remember that a bigger exponent can have a dramatic effect.