# Math Q: Exponents

**URL:** <https://boards.straightdope.com/t/math-q-exponents/338849>\
**Category:** Factual Questions\
**Created:** [January 7, 2006, 4:01am UTC](https://boards.straightdope.com/t/math-q-exponents/338849 "2006-01-07T04:01:33Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\
**Post date:** [January 7, 2006, 4:01am UTC](https://boards.straightdope.com/t/math-q-exponents/338849/1 "2006-01-07T04:01:33Z")

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If 3^2 = 9 and 3^3 = 27, then 3^x = 18.  
How do I find “x” without iterating? Is it possible? Does it have something to do with using base 10 logs and then switching to base 3, perhaps?

(This is not a homework problem, and if you’ve ever seen a homework problem like this, I’d sure be amazed!)

- Jinx

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**Author:** ![Big\_Mony\_Mony](https://avatars.discourse-cdn.com/v4/letter/b/e47774/32.png) [@Big\_Mony\_Mony](https://boards.straightdope.com/u/Big_Mony_Mony)\
**Post date:** [January 7, 2006, 4:10am UTC](https://boards.straightdope.com/t/math-q-exponents/338849/2 "2006-01-07T04:10:59Z")

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You take the log of each side (base doesn’t matter).

log(3^x) = log(18)

Now log(y^x) = x\*log(y) (you can figure this out by fooling around with exponents and logs, but it’s a general rule) so:

x\*log(3) = log(18) or:

x = log(18)/log(3)

x = 2.631 (roughly)

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**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\
**Post date:** [January 7, 2006, 4:16am UTC](https://boards.straightdope.com/t/math-q-exponents/338849/3 "2006-01-07T04:16:44Z")

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Thanks, Big Mony…

But, is it true base does not matter? If I used ln instead of log, the result would be the same? I guess I’ll have to try it out and prove it to myself…

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [January 7, 2006, 4:20am UTC](https://boards.straightdope.com/t/math-q-exponents/338849/4 "2006-01-07T04:20:57Z")

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> [@Jinx](#):
>
> Thanks, Big Mony…
> 
> But, is it true base does not matter? If I used ln instead of log, the result would be the same? I guess I’ll have to try it out and prove it to myself…

log[sub]b/sub = log[sub]n/sub/log[sub]n/sub.

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**Author:** ![Big\_Mony\_Mony](https://avatars.discourse-cdn.com/v4/letter/b/e47774/32.png) [@Big\_Mony\_Mony](https://boards.straightdope.com/u/Big_Mony_Mony)\
**Post date:** [January 7, 2006, 4:28am UTC](https://boards.straightdope.com/t/math-q-exponents/338849/5 "2006-01-07T04:28:25Z")

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Yeah, base doesn’t matter because your doing the same thing with each side. So you could use ln() or whatever, though ln() and base 10 would no doubt be the most common.

Note that logs of different bases very only by a multiplier so:

log(18)/log(3) = 2.631 no matter what base (because the multipliers cancel out).

On preview, what **ultrafilter** said.