# Using Logarithms in the Real World

Logarithms are everywhere. Ever use any of the following phrases?

You're describing numbers in terms of their powers of 10 -- a logarithm. Ever mention an interest rate or rate of return? It's the logarithm of your growth.

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/using-logs-in-the-real-world/

Very tactful Simpsons reference. That’s it - I’m making this my homepage

Your work is TOPS!

I suggest this post and the 2 on the natural log from 2007 be on an easy-use button on your homepage. Also, can you tell me if all logarithms are in scales of 10? Can they be represented in powers related to squares? So that we can figure the log for 57 within the powers of 6 (61, 6*6, etc.)?

I guess most of the people out there don’t even know what the notation log_2 8 means, and even less people know that the result is 3.

For newbies, a good introduction offers this youtube video (in German):

@Joey: Thanks for the comment – glad it came in handy :).

I don’t know who told you we were fed up with logs! It was quite a fascinating unit and I’ve learned a lot from it. I haven’t mastered it, obviously, but I now understand the basic concept. Thanks Miss and thanks Kalid for the slightly different overview of logs!

@Sebastian: Long time! Love it – we have arbitrary distinctions of what “large” means :).

Hi Kalid,

Been a long time. I thought I’d share this little analogy since it seems apros. When thinking of large numbers, I was reminded of the Piranaha Tribe in South America whose number system seems to only go up to 2. Everything else is just “a lot”. That’s how we are…just on a different scale. 1 billion… 1 trillion… what’s the difference? They’re both just bigger than 2

Sebastian

@Audrey: Haha, the coincidences keep popping up! :). Happy it was timely :).

You know, I’m realizing that there are lots of little examples where we use math but because it isn’t labeled as such, we miss it. Over time I’ll probably be adding to this article as new items sneak in!

Kalid, get out of my head! We JUST finished our logs unit, and as happens every year with logs, the kids were so fed up with them… But they were also wondering why they had to learn about them - where in the real world does this come up? When I saw this, I immediately went to our brand new classblog site at http://learnmathclassblogs.blogspot.com/, and posted about your post. I hope this gets you some more comments, and I know your work will once again enlighten anyone who reads it. I love the human-friendly explanation of why we use logs - so simple, so true. I never even thought of that!

This is very informative and should help to show some people that whether you like it or not, maths is everywhere and you can not hide from it.

Incidentally I done a post proving and explaining some of the laws of logarithms: http://www.eloquentmath.com/2012/03/logarithms.html

@Titus: Whoops, I had a typo there (forgot the / 5) and should clarify!

The natural log starts with some growth amount, and gives us the corresponding growth rate (if it happened in 1 period).

So, ln(150/100) = .405, which means “In one period, you can go from 100 to 150 with a 40.5% continuous growth rate”.

However, our growth happened in 5 periods, so we split up that 40.5% growth rate among 5 years: 40.5 / 5 ~ 8% each year.

I actually got lost right at the beginning, unfortunately. It says that: ln(150/100) = 8.1%. How does one get the number 8.1%?

@Paul: Nice, glad you enjoyed the reference

[…] […]

@Shiv: Thanks, I love that video :). Negative numbers are an interesting topic too. A lot of stereo systems show the decibel reading (-10, -20, etc.) and this is an attenuation from the max level (0 = no attenuation, i.e. no decrease in signal).

Another great post Kalid!. Here is a YouTube video that shows the powers of 10. By using order of magnitude you are taken to the edge of the universe and back to a single atom.
Talk about change in perspective!

How about showing the effects of negative numbers in the power?

Hi there, thank you for your page. I have one question:

I take the log of my data and do some calculations with it afterwards (e.g. comparing changes from one year to the other etc.). I use The final results fpr my regression and get no significant results.

If I take the exp after having done my calculations and plug in these new values in my regression I get significant results. I wonder why? Is there any mathematical reason for why the chances of getting significant results with exponents (or having taken exponents after having taken the log) is higher?

Would it make sense to say that taking the exponent from a logged value is like broadening the distances between the values agai and by this - like lense - making them more visible?

Tom

I have enjoyed looking around your site. I can here to see what you had to say on the Rule of 72 and feel it is accessible to high school students (which is what I wanted). While here I looked up what you had to say about logs, and to my surprise you don’t mention John Napier (especially since you said you always wondered what the definition of e was). At any rate I believe Napier to be one of the least respected mathematicians, yet he gave us logs and e (and Napier’s Bones) for very practical reasons. Look him up. And thanks for this site. I will send my students here often.

When I was educated in the early seventies we were told we’d need to provide a slide rule, I remember it didn’t get much use. Found it yesterday in a drawer I was clearing out.

You have not only educated me in a way I didn’t dream of in my teens but renewed an interest in old technology, I’d seen a film documentary which showed a navigator in a Lancaster bomber using one on a night raid over Germany, no doubt calculating time to the next way point.

I’ve worked in IT since 1985 and seen enormous advances in technology, but am amazed at what this gadget can do.

cheers
Martin