The folding function logarithms

Two function are comparable in Linear scales but once the rate of growth becomes exponential then it becomes difficult to compare them.

Logarithms provides a scale to understand algorithm for vert fast changing functions by comparing 1, 10, 100, 1000 range jumps on iteration.

Apart from log base 10, which is commonly used, we also use log base e, ln or natural logarithm. where e = (1+1/n)^n as n reaches infinity. Natural logarithms occurs in many natural events.

Richter scale to measure earth quake is logarithmic.


Please compare the functions and their growth rates. Here log n grows very slowly. It is an mirror image of Exponential, which grows damn faster.

Here are some rules of Logs

Here are some application of rules of logs.

More rules and special names of rules

The products becomes sums, divisions becomes subtraction, exponential becomes multiplication. Apart from these complex operations turning into simple operations. I find even in exponential function, we have same rules.

I find the following log function rules interesting.

1. a ^ logb = b^ log a => an Exchange rule.

2. log a (base b) = log a base c / log b base c => base change rule

3. Logarithm does not work in -ve & - infinity numbers. They stop at 1. Log(1) =0. Log of infinity is infinity.

Tip: Take log of two or more function whenever you find it difficult to compare especially when you have exponential functions. Apply logarithmic rules to simplify comparison.

1/log(x), when we take log becomes log(1) - log(log(x)).

when comparing log with 2 different bases. The log function whose base is larger is the one which has got folded a lot, hence it is smaller than the log function which has a smaller base.