^{20}outcomes of the flipping of twenty coins rather than saying 1.049x10

^{6}ways.

By using that power, your information is likely more accurate. However, this power does not tell you much about the number. Most of us would have no idea if 2

^{20}is in the thousands, millions, billions, etc. at the first glance.
First, lets ask a question. What is the common logarithm of a number, or what can you gather from it? Well, the common logarithm is the power that ten has to be raised to to obtain that number. For instance:

log(100) = 2

log(5000) = 3.69897

log(6283185) = 6.79818

What do you notice about these numbers? It's not clear at first, but count the number of digits in each of the inputs. You will find that the common log is always just a little bit below that number. In fact, to figure out the number of digits in a number, all you have to do is take the common log and round up to the nearest integer.

How can this be used to find the number of digits in a power? Interestingly enough, there is a logarithmic identity stating that the log of a number raised to the power is equal to the power times the log of the number. For example,

log(27) = 3log(3)

ln(32) = 5ln(2)

log(2

^{20}) = 20log(2)
Look at the last example there. We just simplified the gigantic 2

^{20}to a reasonable looking 20log(2), which is the formula to figure out the number of digits it has. In other words, the number of digits in 2^{20}is just 20log(2) rounded to the nearest integer. Plugging this into a calculator tells you that the log is 6.0206, meaning that there are seven digits in the number. If you multiply it out, you will find that 2^{20}= 1048576, which does indeed have seven digits.
So whenever a type of problem pops up with a power of this sort, try to determine how many digits it is. Chances are you will gain a much better understanding of the statistic when you perform this quick calculation.

Hi Ethan,

ReplyDeleteThis is a really great trick! I'm a high school student participating in math competitions, and this looks very helpful. Here's a problem that stumped me on a competition last week:

"Order these numbers from least to greatest: 2^100, 3^57, 10^30, and 2^50 * 5^25."

I hadn't thought of the log trick, and apparently neither did the solution writer: the answer required converting to scientific notation and common factors (I didn't get a good look at the answer sheet, everyone was fighting for it). Thanks again!