hey there… your page has been very informative… i had this project about logarithms which i had to complete in less than a week and your page came in handy… honestly i got most of the info from your post… keep up the good work and thanks again!!!
Hi Clare, glad you liked it! I’d like to make some “getting started” guides for the site, thanks for the suggestion.
Most logarithms are in terms of base 10 (since our regular numbers use that) or base e (since that appears most often in nature). However, you can use any base you wish – programmers often count by twos, so base 2 (or base 8, or base 16) is often used. It comes down to using a convenient number for the types of problems at hand (so base 6 is definitely possible… base 60 is how we count seconds and minutes, for example).
Thanks Charl!
Hi Tom, neat question. I’d have to see the data to make a more specific analysis, but in general, taking exponents (e^x) will expand / broaden the distances between data. This can make gaps more apparent. [Similarly, taking logarithms “shrinks” data onto a smaller scale].
is that the practical applications of logarithm in real life? pardon me. this is part of our project and i’m having trouble with it. if you’re not busy,can you help me? i need 10 practical applications/real life applications of logarithms. thanks a lot.
thanks for the information about logarithm .keep up
How is logarithm used irl? My teacher is offering to boost anyone up an entire letter grade if someone can find how it is used in irl situations.
how would u cite this page?
Hey kalid, thanks for this entire site in general - I’m a huge fan of your work. Your conception of taking e as a scaled version of the final growth product and ln as the amount of time has helped tremendously - on an intuitive level, how do these metaphors intuit the law that says the log of a power of a number is the exponent times the logarithm of the number? This is one of those rules that I memorized in high school, but i’m having a hard time seeing it with the metaphors that you have offered.
Thanks again!
Very good
Hey Alex, really glad it’s helping. Check out http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/ for more discussion on the laws of logs and how multiplication/powers work.
At its heart, something like 3^4 means “take 3x growth, then wait and grow 3x again, then wait and grow 3x again, then wait and grow 3x again.”
How long should that entire process take? If the time to grow a single 3x is log(3), then the time to grow 3x four times should be:
log(3) + log(3) + log(3) + log(3) = 4 * log(3)
And we can check our intuition and see that log(3^4) = 4 * log(3). Check out the post for more details/examples.
Hey Khalid,
I must say you have some great work here! They ease with which you explain these topics quite astound me. 
I do have a rather silly question, though. In your 6-Figure numbers example, the log of 500,000 comes to 5.7. Why did you add the extra 1?
Hi Tanvi, great question, glad you’re enjoying it :).
Logarithms count the number of multiplications needed to reach a number from a certain base (like 10).
For example, to get from 1 to 100 in base 10, we need two multiplications: 1 x 10 x 10 = 100.
However, the number of digits in 100 is clearly 3.
This difference (number of multiplications vs. number of digits) means the digit count is different from the logarithm.
The reason for this difference: we start with a single digit (1) and each multiplication by 10 gives us another digit. The logarithm is a count of the “extra multiplications we added in” but not our starting point.
When looking at 500,000, we can say the logarithm is 5.7. But how many digits does it have? Six – and I’d even describe it as a “6.7” digit number to distinguish it from a number like 100,000 which is exactly 6 digits.
Oh I got it now. Thanks Khalid.
Forty years after my last maths lesson, I now find I have to get up to speed on logarithms. Excellent post, clear and well written, it brought it all back to me in a few minutes.
Hi Kalid,
Enjoyed this very informative post. Can you also write about electricity voltage, current etc.
Nilesh Joglekar
Thanks David, happy to hear it clicked.
100 increases to $150 in 5 years. What is the yearly interest rate? I got 5√150/100=8.45%. and when working it out logically, that’s the right answer. How did you get ln (150/100) /5=8.12%?
Thanks for this post. I’m trying to understand how Moz Domain Authority (DA) works. They say DA is measured on a 100-point logarithmic scale.
Does that mean DA30 is 10x more powerful than DA20?
OR
Is DA21 10x more powerful than DA20, and DA30 would then be… 10^10x (100 million times)?
OR
Is it base-2 not base-10 and DA21 is 2x more powerful than DA20, but DA30 would be 1024x more powerful than DA20…
Help?!?! 
Thanks,
Josh
@Andrew: Great question. 8.45% is the yearly interest rate, assuming we compound every year. 8.12% is the yearly interest rate if we assume we can compound continuously.
If we compound continuously, we’ll end up with 8.45% at the end of each, but an individual dollar only expects to gain 8.12%. It’s the “interest that interest years” that boosts the rate up to 8.45%. e and natural log deal with the perspective of what an individual dollar thinks it is earning. There’s more here: http://betterexplained.com/articles/think-with-exponents/
@Josh: Good question. 100-point logarithmic scale could be setup in any way, but I’m assuming they mean base 10, and that every 10 points means 10x more powerful.
(Having every point be 10x more powerful isn’t necessary, since 10^100, the max of the scale, is an enormous number. Also, having a base 2 scale isn’t that human friendly.)
In this case, every 10 points would be a 10x increase in importance (10, 20, 30, 40, 50) would be (10, 100, 1000, 10k, 100k).
Just my guess but that’s similar to how page rank works (except it’s a 1-10 scale vs 1 to 100, so 1, 2, 3, 4, 5 is 10, 100, 1000, 10k, 100k).