Demystifying the Natural Logarithm (ln)

Hi Mimi, thanks for the wonderful comment! I’m glad you found it helpful – I agree, there are many subjects that are often explained in an overly complex way. Everything can (and should) be simple :).

Do all bases for logs have to be positive? If not is the log of a negative number possible?
EG -8 = (-2)^3
Is this 'log to base -2, of -8 = 3
Help please and thanks. JP

Hi Jack, great question. At its heart, a logarithm answers the question

base^power = desired value

So I don’t see a reason why you couldn’t use a negative base. The only problem is that certain values are undefined: with regular, positive bases we can’t get to negative values (at least, not without using imaginary numbers).

With negative bases, certain values are out of range: for example, we can’t get “-4” since (-2)^2 = 4.

So, negative bases would work, but would have a different set of undefined values.

I really like the way you explain about e,you make a really good teacher

Thanks Doug, glad you liked it! :slight_smile:

Excellent explanation…logical, applicable, and concise. I teach nuclear plant operators a variety of technical subjects/skills (including mathematics), and this is the type of format I strive for in my lessons. Less memorization and more in-depth understanding means more retention! I am going to recommend your site to both my students and my fellow instructors.

Wow, thanks Tim, that’s great! I love hearing that people in the real world are finding the content useful.

:slight_smile: :slight_smile: :slight_smile:
The world make’s sense again - you just reminded me why i like math…
thank you!

Awesome Katie, glad you enjoyed it!

Re: comment #41: saying that “certain values are undefined” is IMO somewhat misleading. If you would allow negative bases, such as -2, you would get a very ‘messy’ function.

If you realise the log is the inverse of an exponential function with the same base, allowing -2log(x) would be the same as allowing -2 as the base in an exponential function: y = (-2)^x.
Now this last function is only defined for integer x: (-2)^0.5 for example is asking for the square root of -2 which clearly is undefined (assuming non-complex numbers). So the graph of (-2)^x consists of dots which are horizontally spaced 1 apart (alternating above and below the x-axis: the sequence (-2)^1, (-2)^2, (-2)^3 is -2, +4, -8, +16, etc).
Likewise the graph of log(x), base -2, would consist of dots vertically spaced 1 apart and alternating left and right of the y-axis (-2log(-2) = 1, -2log(4) = 2, -2log(-8)=3, etc) – the reflection of (-2)^x in the line y=x.

So yes, you’re right that “certain values are out of range”. The problem though is that almost all values are out of range, seriously limiting the usefulness of a log function with negative base. There’s a good reason mathematical practice explicitly limits the choice of a base for log and exponential functions to positive numbers (excluding 1, for obvious reasons).

Your remark that “I don’t see a reason why you couldn’t use a negative base” seems to me to add confusion rather than clear things up.

Somewhat related observation: when dealing with discrete sequences, all the above is not a problem. You can easily define a sequence x(0) = 1, x(n) = -2*x(n-1), giving the sequence 1, -2, 4, -8, … This demonstrates how such a sequence resembles but is not identical to exponential growth.

Don’t take my comments to be negative though: I really enjoy your website, including the above article.

Hi Hendrik, thanks for the comment! No worries, lively discussion is one of the fun parts of having a blog. Nobody has all the answers or is immune from mistakes :).

Yes, looking at it again I should have clarified my response to the negative base. As you say, while it is “possible” to have a negative base, it’s likely not very useful because it only works for a small set of numbers. (As an aside, a negative base may give an interesting graph if you trace out the complex numbers it covers). Appreciate the comment.

Your detailed content really shine. Yours is one of the few that has an in-depth explanation. Maths is, at times, abstract and takes time to figure out the concepts. Here I find it smooth going. Though I have maths blog myself, I like your blog. Good work! Nice learning from you. Natural log and time… cheers!

your presentation is very good. it has a practical
insight, but when you come up with 2^n for the doubling equation you seem to pull that out of the hat. for sombody familiar with math its easy to see why but for sombody unfamiliar its mysterious. how did you get there? then you pull out the 100% again how did you get there? then you use the 2^n equation for all other forms. i understand why but i could never derive this and wonder that there is still somthing i dont get.

@Lim Ee Hai: Thanks for the wonderful comment! I’m glad you’re finding the site useful. Yes, math has many beautiful ideas, but they can get hidden behind a wall of abstraction, or lost in a sea of formulas. Glad you’re enjoying it.

@Mike: Great question, it’s always helpful to see what parts worked and what didn’t. There’s more about the 2^n equation in the article on e, but I’ll try to explain it here.

Doubling can be considered as growing 100% – if you start the year with 1 item, you finish with 2. So 2 = (1 + 100%).

If you double 3 times, you could write that as 2 * 2 * 2. That is, after 3 years, you would have 8 items.

But since exponents are a shortcut for repeated multiplication, we can write 2^3 instead of 2 * 2 * 2. Even better, we can pick any number of years and write


where n is the number of years to use. Remembering that 2 implies a rate of 100% increase, we could replace that with

(1 + rate)^n

to get a general equation for any rate and duration. You may also want to try this article on interest rates.

Hope this helps!

[…] why am i goin about with this, i know not. and that pretty pinky graph over there, thats just some random stuff i got from reading about the natural log. actually i was stumped at integrating some weird ln (some go “lon”, some go “el, en” but i so love the “lin” pronounced ones. like, my name! yeah!) stuff so i go wiki, and as usual i ended up with soemthing else. nyway, the lecturer was speaking about how natural log arises, with its weird form and all, and why dont we just do simple stuff like log ten to model things by. apparently most of the natural occurrences can be described mathematically by the exponential law, and subsequently the natural log. beautiful stuff, i must admit. lost da link, but this one’s not bad. […]

[…] A premature focus on rigor dissuades students and makes math hard to learn. Case in point: e is technically defined by a limit, but the intuition of growth is how it was discovered. The natural log can be seen as an integral, or the time needed to grow. Which explanations help beginners more? […]

I am going to create my own world now!

Kalid -

Great explanation and very helpful. Thanks for taking the time to put it together.


Hi David, appreciate the comment – I’m glad the explanation worked for you!

Thank-you! I have a degree in math I have wondered for years why e was used. In all of my classes I was told what e equaled, that ln was the inverse of the e function, and how to compute with e and ln, but never how e was derived! I now have a better understanding and hopefully this will make me a better math teacher in the future! :slight_smile: