Math CentralQuandaries & Queries


Question from kishor, a student:

whats the easy way to calculate X raised to Y where y is much greater than x.

Hello Kishor.

Most people ask this kind of question when dealing with very large numbers their calculators can't handle.

I'll show you with an example:

What is the value of 34.11722.3 ?

Clearly, many calculators would balk at this number. Another way to calculate it in order to express it in scientific notation would be to use logarithms.

The useful logarithm rule is this:

Given a = bc, we can take the logarithm in any base d on both sides of this equation and get:

logd a = logd bc = c logd b.

Since scientific notation uses base ten (decimal), we want d = 10.

So for our example, we have:

Let a = 34.11722.3. Then

log10 a = 1722.3 (log10 34.1) = 1722.3(1.532754379) = 2639.8629.

So a = 102639.8629. Now we break it up.

a = 102639.8629 = 10(0.8629 + 2639) = 100.8629 × 102639

a = 7.29 × 102639.

That's the method I use. Here's the short form for nm:

  1. Calculate p = m log10 n.
  2. Break up p into its whole number (w) and its decimal (d) portions.
  3. Calculate f = 10d.
  4. The answer expressed in scientific notation is nm = f × 10w.

Hope this helps,
Stephen La Rocque.

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