# log Base 10

#### SamR

Joined Mar 19, 2019
4,614
It's been a while since I've used logarithms but doesn't the unannotated term log imply base 10? So that when using base 2 or base e then it would need to annotated as such?

#### xox

Joined Sep 8, 2017
800
From a "pure" mathematical perspective, "log" means the natural logarithm. In the sciences (and elsewhere), log is instead used to mean the base ten logarithm, although certainly not always. Boltzmann's entropy formula for example uses it in the former sense. Earthquakes, on the other hand, are measured on a log10 scale, and yet sometimes referred to in various literature as simply being "logarithmic". So it really just depends on the context where it appears.

#### hrs

Joined Jun 13, 2014
369
In LibreOffice spreadsheet log() is log10. Excel too I think. In Python log() is log e.

#### crutschow

Joined Mar 14, 2008
32,064
Earthquakes, on the other hand, are measured on a log10 scale, and yet sometimes referred to in various literature as simply being "logarithmic".
Whether it's measured in base 10 or base e, makes no difference to the designation as being "logarithmic".

#### WBahn

Joined Mar 31, 2012
28,492
It's been a while since I've used logarithms but doesn't the unannotated term log imply base 10? So that when using base 2 or base e then it would need to annotated as such?
Depends on the context, so it's safest to use a subscript with the log function and to check, when possible, what base the log is being taken to. Most programming languages use log() for the natural log and log10() for the common log.

#### SamR

Joined Mar 19, 2019
4,614
K, thanks guys! Decades ago when using my slide rule, it was log or ln. Or at least that is sorta how I remember it. Not a bad suggestion to subscript it to remove any ambiguity.

#### MrSalts

Joined Apr 2, 2020
2,767
Best way to know is to try it.
If you ask someone, you'll get many opinions and still not know.

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#### MrChips

Joined Oct 2, 2009
28,515
It depends on what platform you use.
In MATLAB, log10( ) is log base 10 and log( ) is natural log.

#### SamR

Joined Mar 19, 2019
4,614
Using MS Mathematics and my various TI calculators. Actually seems like my TI-36 PRO works the best for logs except I am unable to set the base on any of them it seems. The 36 has a quirk in that it somehow cannot solve the square root of 3? The 83 you have to scroll through a catalog to find the log function so it is a PIA. The nspire is a poor substitute for a computer so I use my computer with MS Math instead. Haven't read the help file for MS Math to see what all it can do with log.

Edit: Yep, for MS Math log(x) is base 10 but for log(b,x) b is the base so that's good news.

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#### MrSalts

Joined Apr 2, 2020
2,767
Using MS Mathematics and my various TI calculators. Actually seems like my TI-36 PRO works the best for logs except I am unable to set the base on any of them it seems. The 36 has a quirk in that it somehow cannot solve the square root of 3? The 83 you have to scroll through a catalog to find the log function so it is a PIA. The nspire is a poor substitute for a computer so I use my computer with MS Math instead. Haven't read the help file for MS Math to see what all it can do with log.
MSExcel also works well.
The NSpire also has a TI-84 faceplate that can turn it into a TI-84 (or maybe that was a special edition Nspire. But the TI-84 is a great calculator (for me).

#### MrChips

Joined Oct 2, 2009
28,515
As long as you have any log( ) function it is easy to convert from one base to another.

#### SamR

Joined Mar 19, 2019
4,614
Nope, I misspoke. It is the Nspire that you have to dig through their catalog to find the Log function. The 83 has log and ln on the top keys so you don't even have to shift to get to them. And it does know how to do square root of 3 unlike their 36! But the 36 doesn't use batteries and works in low light conditions plus a bit smaller and a lot less heavy.

Edit: Did a bit more digging on the Nspire. Log is NOT covered in the TI guidebook but I found out that it is easy to get to the log functions and it DOES support bases. Might actually be easier than using MS Math.

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#### WBahn

Joined Mar 31, 2012
28,492
K, thanks guys! Decades ago when using my slide rule, it was log or ln. Or at least that is sorta how I remember it. Not a bad suggestion to subscript it to remove any ambiguity.
Slide rules were set up with base-10 in mind because they were tools to facilitate doing base-10 arithmetic.

Most scientific calculators have as their primary audience high school and college students. Most high school students (historically, at least) get introduced to logs right after they learn about exponents and radicals with the application being as a means of doing multiplication and division using addition and subtraction, so they are learning it in the context of base-10 operations, likely two or three years before they ever get to the point of learning about the existence of natural logs (which are often introduced when they are learning integral calculus).

Consequently, it is natural (no pun intended) for them to think of log() as being the base-10 logarithm and the calculators that they buy sell better if the notation on them is in line with this.

This is one of any number of examples that shows that the professional side of a field (working mathematicians, for whom log() means natural log by default) and the educational side (where log() means the common log by default) are different critters that are not always in sync.

#### crutschow

Joined Mar 14, 2008
32,064
My old HP42s will also do the square-root of -3 (or any negative number) as it can handle complex numbers.

It uses LOG for base 10 and LN for base e logarithms.

(The picture is from my virtual 42s on my laptop.)

#### SamR

Joined Mar 19, 2019
4,614
Once I figured out how to use the Nspire, if I don't enter a base number in the subscript space, it assumes base 10 and fills in the subscript as 10 automatically. In an older text I was perusing today it mentioned using lg for base ten instead of log. Maybe it's old age but I don't remember anything about a base other than 10 or e. Once I got the hang of using base numbers and the theory behind it, it's not so bad and makes a lot of sense. So I'm surprised I don't remember ever seeing it before. It makes it pretty easy to convert from log to exponential expressions.

Also discovered that the Nspire won't resolve the square root of 2. Very strange, the 36 won't do 3 and the Nspire won't do 2?

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#### WBahn

Joined Mar 31, 2012
28,492
Once I figured out how to use the Nspire, if I don't enter a base number in the subscript space, it assumes base 10 and fills in the subscript as 10 automatically. In an older text I was perusing today it mentioned using lg for base ten instead of log. Maybe it's old age but I don't remember anything about a base other than 10 or e. Once I got the hang of using base numbers and the theory behind it, it's not so bad and makes a lot of sense. So I'm surprised I don't remember ever seeing it before. It makes it pretty easy to convert from log to exponential expressions.

Also discovered that the Nspire won't resolve the square root of 2. Very strange, the 36 won't do 3 and the Nspire won't do 2?
In several fields, lg() is used to indicate the base-2 logarithm. But this just underscores that it is unsafe to make assumptions about what base is being meant (except that ln() is all-but-guaranteed to mean natural log) -- different people in different fields mean different things. So, when YOU write something or document something, be explicit. If nothing else, put a note somewhere that indicated what your nomenclature is.

I can pretty much guarantee that those calculators will have not a problem calculating the square root of either 2 or 3. Either you are doing something wrong, which is by far the most likely explanation, or your particular calculators have something wrong with them. It's possible, but unlikely, that two calculators have a similar problem. Review the documentation to see if you are overlooking something subtle. If not, then you might put in new batteries (if it uses them).

#### SamR

Joined Mar 19, 2019
4,614
Batteries are not the problem. Found out entirely by accident during a calculation. In both cases it shows the root symbol with the number under it and when enter is pushed it moves from the left-hand entry side of the screen to the right-side solution side. The older TI-83 has no problem with either. I've done a lot of root calculations over the years but nothing simple like finding the root of 2 or 3 until I started doing the exponential equivalents of some Log to different roots expressions. If I multiply or divide it, it remains as a symbol. But if multiply 2 of them against each other it solves or if I raise the square root symbol to any power it solves. Nothing that I've read in their literature about it.

Also the 36 does do 3^1/2 so it can express the square root of 3. Such a weird "feature"...

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#### bogosort

Joined Sep 24, 2011
696
Batteries are not the problem. Found out entirely by accident during a calculation. In both cases it shows the root symbol with the number under it and when enter is pushed it moves from the left-hand entry side of the screen to the right-side solution side. The older TI-83 has no problem with either. I've done a lot of root calculations over the years but nothing simple like finding the root of 2 or 3 until I started doing the exponential equivalents of some Log to different roots expressions. If I multiply or divide it, it remains as a symbol. But if multiply 2 of them against each other it solves or if I raise the square root symbol to any power it solves. Nothing that I've read in their literature about it.

Also the 36 does do 3^1/2 so it can express the square root of 3. Such a weird "feature"...
I have a Ti-36X Pro, which does the same thing. The calculator defaults to showing you the exact answer (if possible) when pressing the 'enter' key, which can be handy (e.g., typing $$\sqrt{18}$$ and pressing 'enter' will result in $$3 \sqrt{2}$$, and typing $$\frac{17}{51}$$ results in $$\frac{1}{3}$$), though this isn't always desired, as you've found. Press the '$$\approx$$' key above the 'enter' key to get the approximate decimal representation.

Strangely, the situation is reversed when using the built-in functions: typing $$\sin^{-1}(1)$$ and pressing 'enter' gives you 1.570796...; pressing $$\approx$$ gives you the exact result $$\frac{\pi}{2}$$.