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# How come 3^(221/92) is very very nearly fourteen?

Is there supposed to be some deeper mathematical meaning in this?

### 12 Answers

- PinkgreenLv 71 month ago
Let y=3^(221/92), then

log(10)y=(221/92)log(10)3

=>

log(10)y=2.402173913(0.477121254)

=>

log(10)y=1.146128223

=>

y=10^1.146128223

=>

y=14.00000604~14.

- AmyLv 71 month ago
Because you typed log base 3 of 14 into your calculator and asked it to display the result as a fraction, and the answer was 221/92.

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- Bill-MLv 71 month ago
There is nothing special about that number.

221 divided by 92 is not a special number. 2.4022

- SlowfingerLv 61 month ago
No deeper meaning.

221/92 is just a fairly good approximation of logarithm

log₃(14) = log(14) / log(3) = 2.4021735027...

which is a transcendental number

It is always possible to find fractions that approximate transcendental numbers with increasing accuracy.

For example

3^(12/5) = 13.9666101652...

more accurate is

3^(221/92) = 14.0000063108...

but

3^(572719/238417)= 13.99999999993129....

How many decimal places do you need?

- PuzzlingLv 71 month ago
Let's assume you wanted an exponent that would make it *exactly* 14:

3^x = 14

We could take the common log of both sides:log(3^x) = log(14)

Then use this rule of logarithms --> log(a^b) = b log(a):

x log(3) = log(14)

x = log(14) / log(3)

x ≈ 2.40217350273...

Let's look at the value of your fraction:

221/92 = 2.40217391304...

You've picked a fraction that is the same out to 6 decimal places, so it makes sense that the result would be pretty close to 14.

There is no deeper meaning other than you've gotten a decent rational approximation of log(14) / log(3).

You can get a continued fraction form of log(14) / log(3) as:

x = 2 + 1 / (2 + 1 / (2 + 1/(18 + ...)))

If we stop the continued fraction after 18, we get 221/92 which is what you had.

- DavidLv 71 month ago
3^221/92 is the same as 3 to the power of 221/92 or 92 root of 3 to the power of 221 and they both work out as 14.00000631 or 14 rounded to nearest integer which means a whole number.