Can a log have multiple bases?

[YouAreAwesome]

TL;DR Summary

Does it make any sense to say: ln to the base 2?

Hi,

I tutor maths to High School students.

I had a question today that I was unsure of. Can the natural log be to the base 2?

The student brought the question to me from their maths exam where the question was: Differentiate ln(base2) x^2

If the natural log is the inverse of e then how does the natural log to the base 2 make any sense?

Thanks for any help on this.

YAA

 

[DrClaude]

What is meant by "natural log" is $\log_\mathrm{e}$, so no, you can't have "natural log base 2". What you would have is $\log_2$.

 

[YouAreAwesome]

What is meant by "natural log" is $\log_\mathrm{e}$, so no, you can't have "natural log base 2". What you would have is $\log_2$.

Thank you, that is what I thought. Can (Log_e)₂ ever make sense? The question was written by a math teacher who is apparently defending the question and claims to have a solution. I told my student to please bring me a copy of that solution. It will be interesting to see what he comes up with. I will post here if it is of interest.

 

[FactChecker]

Thank you, that is what I thought. Can (Log_e)₂ ever make sense?

I would interpret $(...)_2$ as a number written in binary, which is something that you can do with any number. It has nothing special to do with logarithms. So $(log_e(...))_2$ makes sense to me.

 

[fresh_42]

I thought about $\log_2(\log_e)$ but this would lead into trouble for values less than $1$. Maybe he meant $\log_{2e},\log_{e^2},\log_{2+e}$ or $\log_{\log_2e}.$ The logarithm solves $a^x=b.$ There is only one $a.$

 

[jbriggs444]

I would interpret $(...)_2$ as a number written in binary, which is something that you can do with any number.

To be more precise, you can apply a "2" subscript to a digit string (with optional radix point) in order to designate it as a base 2 numeral rather than the default of a base ten numeral.

Applying the same subscript to a number has no such meaning.

The way that I would express "the base two expansion of the natural log of x" would be with those words rather than by saying: $(\ln{x})_2$.

So I could say that $\ln(10) \approx 10.01001101011101100011010111100111_2$

Or that $2.302581787109375_{10} \approx 10.01001101011101100011010111100111_2$

But I could not say that ${\ln(10)}_2$ = "10.01001101011101100011010111100111...".

 

[Mark44]

I would interpret (...)₂ as a number written in binary

That would be a reasonable interpretation normally, but IMO not in the context of this thread.

Summary: Does it make any sense to say: ln to the base 2?

If the natural log is the inverse of e then how does the natural log to the base 2 make any sense?

In the above I assume that the OP is using "base" in the context of "log base e" or "log base 2" ($\log_e() = \ln()$ or $\log_2()$) and so on. I'm reasonably certain that "base" here doesn't mean the number base (binary, decimal, hexadecimal, etc. representation of a number).

 

[YouAreAwesome]

That would be a reasonable interpretation normally, but IMO not in the context of this thread.

In the above I assume that the OP is using "base" in the context of "log base e" or "log base 2" ($\log_e() = \ln()$ or $\log_2()$) and so on. I'm reasonably certain that "base" here doesn't mean the number base (binary, decimal, hexadecimal, etc. representation of a number).

That's right. The exam is an Australian NSW HSC exam created by the school. Binary/hexadecimal etc is not included in the NSW syllabus and therefore can not be examined or tested. $(\log_e)_2$ means "log to the base e to the base 2" where "to the base e" and "to the base 2" have identical meanings with regard to "to the base".

 

[FactChecker]

$(\log_e)_2$ means "log to the base e to the base 2" where "to the base e" and "to the base 2" have identical meanings with regard to "to the base".

I don't know what "log to the base e to the base 2" means. If it means "log to the base 2 of log to the base e" then I can understand it. But then why isn't it written $log_2(log_e( ))$? That seems much more standard and absolutely well-defined.

 

[olivermsun]

I think YouAreAwesome is saying the "to the base e" and "to the base 2" play the "same parts of speech" in the expression and hence conflict with one another. At least that's how I read it.

 

[FactChecker]

To be more precise, you can apply a "2" subscript to a digit string (with optional radix point) in order to designate it as a base 2 numeral rather than the default of a base ten numeral.

Applying the same subscript to a number has no such meaning.

Why?

The way that I would express "the base two expansion of the natural log of x" would be with those words rather than by saying: $(\ln{x})_2$.

So I could say that $\ln(10) \approx 10.01001101011101100011010111100111_2$

Or that $2.302581787109375_{10} \approx 10.01001101011101100011010111100111_2$

But I could not say that ${\ln(10)}_2$ = "10.01001101011101100011010111100111...".

Why? I would say that and think that it has a valid meaning.

 

[jbriggs444]

Why?

Because a number and a numeral are different things.

Why? I would say that and think that it has a valid meaning.

I disagree strongly. There is no well-understood convention that putting a numeric subscript to the right of an expression means "convert this numeric result into a text string consisting of its place-value expansion in the given base".

There is a well understood convention that putting a numeric subscript to the right of a digit string means "interpret this digit string using place-value notation in the given base".

The two meanings are exactly the opposite of one another. The one is a conversion from number to text. The other is a conversion from text to number.

 

[FactChecker]

Because a number and a numeral are different things.

I disagree strongly. There is no well-understood convention that putting a numeric subscript to the right of an expression means "convert this numeric result into a text string consisting of its place-value expansion in the given base".

There is a well understood convention that putting a numeric subscript to the right of a digit string means "interpret this digit string using place-value notation in the given base".

The two meanings are exactly the opposite of one another. The one is a conversion from number to text. The other is a conversion from text to number.

Maybe. I think that I would be fine with it. I don't see a problem.

 

[jbriggs444]

Maybe. I think that I would be fine with it.

One does not usually find mathematicians wanting to talk about the expansion of a number in a particular base. It is not relevant to the mathematics. It seems a waste of notation to devote a syntax to a semantic that is rarely used.

 

[fresh_42]

It seems a waste of notation to devote a syntax to a semantic that is rarely used.

Here is a list of notations that are used for binary numbers. No convention at all.

${}_210111$
$[10111]_2$
$10111_2$
$10111(2)$
$0b10111$
$\%10111$ (Motorola-Konvention, also e.g. DR-DOS DEBUG)
$HLHHH$
$L0LLL$
$0111b$ (not recommended for confusion with hex numbers)
$10111B$ (not recommended for confusion with hex numbers)

 

[olivermsun]

One does not usually find mathematicians wanting to talk about the expansion of a number in a particular base. It is not relevant to the mathematics. It seems a waste of notation to devote a syntax to a semantic that is rarely used.

Most research papers, lecture notes or even scribbles on a blackboard (or napkin) that I've seen will, after dealing with some particular thing more than a couple or three times, either adopt some existing notation or invent a new one purely out of convenience. (See for example fresh_42's list of binary notations above.)

Also I am not sure why you think expansion of numbers in particular bases is irrelevant to the mathematics. What if the mathematics in question are about the numerical representations?

 

[jbriggs444]

Most research papers, lecture notes or even scribbles on a blackboard (or napkin) that I've seen will, after dealing with some particular thing more than a couple or three times, either adopt some existing notation or invent a new one purely out of convenience. (See for example fresh_42's list of binary notations above.)

All of those are notations for interpreting numerals. None are notations for converting numbers to text.

But yes, nothing prevents one from adopting a particular notation when it is convenient to do so. One would explain the notation before use in such a case. It is poor manners to invent a notation and then use it without warning.

Also I am not sure why you think expansion of numbers in particular bases is irrelevant to the mathematics. What if the mathematics in question are about the numerical representation

Extremely uncommon. Most mathematics that involves numbers at all is about the numbers, not their representations.

Not non-existent, of course. There is, for instance, the Baily Borwein Plouffe formula to compute the digits for pi in hexadecimal. Still, I do not think that the hexadecimal expansion of $\pi$ is typically referenced as $\pi_{16}$. For instance, the article linked above never uses such a notation.

 

[olivermsun]

Most mathematics that involves numbers at all is about the numbers, not their representations.

I dunno. I'd argue that the entire field of numerical analysis is quite interested in representations.

 

[jbriggs444]

I dunno. I'd argue that the entire field of numerical analysis is quite interested in representations.

So in numerical analysis, what notation is most frequently used to refer to the 64 bit IEEE floating point model number nearest to a given real number $x$?

 

[olivermsun]

No idea what is the "most frequently used" notation for that particular function.

How is that relevant to whether the representation of the number is important in numerical analysis?

 

[jbriggs444]

No idea what is the "most frequently used" notation for that particular function.

How is that relevant to whether the representation of the number is important in numerical analysis?

We are discussing where there is an accepted and agreed-upon notation for the radix expansion of an arbitrary expression and, more specifically, whether the notation consists of a subscripted decimal number to the right of the expression, which number is called the "base".

You are claiming that this is so. I am claiming that it is not so.

As part of the argument, I suggested that such notation would not be widely useful since most mathematics is unconcerned with the representations of numbers. You then suggested numerical analysis is a branch of mathematics where representations of numbers are important, ignoring the fact that numerical analysis is only a small corner of mathematics, but were unable to point to a use of any such notation within the field of numerical analysis.

I would actually argue that actual representations of numbers are less important in numerical analysis than various attributes of the coding scheme that is employed. How many code points? How densely do they fit areas within the range? What range is covered? How uniform is the coverage? Given a particular relative error for numbers in a particular range, how can we minimize the relative error in the result of a calculation? A subscript notation for "the representation of x" is not needed to be able to express these questions or to work on the answers.

 

[olivermsun]

We are discussing where there is an accepted and agreed-upon notation for the radix expansion of a number. You are claiming that there is. I am claiming that there is not.

Actually I did not claim that at all. (Go back and read the thread.)

 

[jbriggs444]

Actually I did not claim that at all. (Go back and read the thread.)

Fair enough. So can we agree that numerical analysis is irrelevant to this thread then and that the existence of that branch of mathematics is not in conflict with any claim that I have made?

 

[olivermsun]

Fair enough. So can we agree that numerical analysis is irrelevant to this thread then and that the existence of that branch of mathematics is not in conflict with any claim that I have made?

If we can agree that your claims are irrelevant to this thread, then I agree that any conflict with your claims is also irrelevant.

What is more relevant is what did the teacher mean when writing the problem? And what's the claimed solution?

 

[YouAreAwesome]

Here is a small snippet from the teachers solution:

$ln_2 x^2 = 2ln_2 x = 2lnx/ln2 = 2/ln2 * lnx =...$

I kid you not.

If you have a simple way to show why this is false $2ln_2 x = 2lnx/ln2$ I would appreciate your contribution.

My strategy is to argue that $ln_2 x$ is incoherent, but perhaps there is an example that can show this more clearly.

 

[olivermsun]

Based on the solution snippet, your and Dr. Claude's first reaction seems right. There is some confusion about the notation $\ln{x}$, which already implies "log to the base $e$", making the subscript in $\ln_e{x}$ redundant.

However if you can overlook the weird notation and pretend that it says $\log$ everywhere it says $\ln$, then
$$\log_2{x^2} = 2 \log_2{x} = 2 \log{x} / \log{2} = 2 / \log{2} \cdot \log{x} = \ldots$$
actually looks fine.

 

[Mark44]

My strategy is to argue that $ln_2 x$ is incoherent

I agree with your assertion that $ln_2 x$ is incoherent, and agree with @olivermsun that perhaps the instructor really meant $log_2$ instead of the unfortunate mess of $ln_2$.

 

[ago01]

Here is a small snippet from the teachers solution:

$ln_2 x^2 = 2ln_2 x = 2lnx/ln2 = 2/ln2 * lnx =...$

I kid you not.

If you have a simple way to show why this is false $2ln_2 x = 2lnx/ln2$ I would appreciate your contribution.

My strategy is to argue that $ln_2 x$ is incoherent, but perhaps there is an example that can show this more clearly.

That seems like a teacher hastily writing something on a board and a better question for the teacher to answer. Logs are all related and it would strike me the teacher meant log base 2 and not ln base 2. ln base 2 makes no sense.

 

[swampwiz]

No, the natural log is the log to the base to the glorious number known as e. It's unfortunate that one must go through calculus (and beyond) to appreciate e (I consider it more important than π).

 

[YouAreAwesome]

I agree with your assertion that $ln_2 x$ is incoherent, and agree with @olivermsun that perhaps the instructor really meant $log_2$ instead of the unfortunate mess of $ln_2$.

This was my thought also, that he simply made a typo and meant log rather than ln, but he has apparently dug his heals in attempting to teach my student why $ln_2$ is accepted. I have sent her to the Head of Maths at the school and we'll see if the problem lies at the root or not.

 

[YouAreAwesome]

Ok, all three mathematics teachers at the school (including the head of maths) AND the exam itself's formal marking guidelines (which were provided from an external supplier) all agreed this was legal notation:

$ln_2=log_2e$ or in words, "log to the base 2e".

Has anyone ever seen this notation before?

 

[Motore]

I guess they mean it like this: $$\ln_2 = \log_{2e}$$, because this doesn't make sense: $$\ln_2 = \log_2 (e)$$

Then perhaps it is valid, but I myself never saw such notation.

 

[YouAreAwesome]

I guess they mean it like this: $$\ln_2 = \log_{2e}$$, because this doesn't make sense: $$\ln_2 = \log_2 (e)$$

Then perhaps it is valid, but I myself never saw such notation.

Yes, $$\ln_2 = \log_{2e}$$
I have not seen this notation either. And I wonder if it is "legal" to test 17 year old students with such ridiculous notation that has not been taught to them in class, or given as required learning in any sense!

 

[DrClaude]

I've never seen it before. I checked in the Oxford User's Guide to Mathematics, and it is not mentioned in the discussion of logarithms. It is up to them to give citations to references where this is presented.

 

[olivermsun]

I’ve never seen this notation, but to dispute its fairness I suppose you’d have to establish whether the textbook or study material introduced the notation.

Accepting the notation for the moment, how does this change one’s interpretation of the solution snippet posted earlier?