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# Logarithm

For computer logs see logging.

Logarithm (from Greek "logo arithmos", roughly "ratio/word number", often shortened to log, lg or ln) is a very important mathematical function telling us to what power a number has to be raised to in order to obtain another number; i.e. in the language of mathematics it is the inverse function to exponentiation. Logarithms are enormously important and useful: for example they allow fast and efficient multiplication and division of numbers (see e.g. sliding rule), solving certain kinds of equations but also, very notably, they introduce logarithmic scales -- a type of alternative measuring scales (to the traditional linear scales found on rulers etc.) in which steps are not spaced by constant distance but by constant ratio, and it turns out this is exceptionally handy for measuring and plotting various values (such as loudness, earthquake intensity etc.). This is because in nature it often happens that the step to "the next level" is not an additive constant, but rather some ratio of previous step (or imagine a video game where leveling up each subsequent level takes more and more experience, let's say three halves of that required for the previous one). So dealing with logarithms sort of takes us from the realm of additions and differences to one where multiplications and ratios rule.

Human senses are known to often perceive logarithmically (and this is exploited in lossy compression algorithms) -- one of the best examples are musical tones: what we hear as an increase in pitch by one semitone is actually a frequency that's 12th root of 2 TIMES increased frequency of the previous one. Our sensation of time passing or visual perception of brightness is also similarly non-linear. Interestingly some studies even suggested that "logarithmic thinking" is possibly more natural to our brains and only at school we're forced to adopt the traditional "linear thinking". However it might be, the first confrontation (normally during high school) with the mathematics of logarithms usually scares people off and average IQs never fully grasp them (because they "don't need it in real life"), but they're really not hard, just require some getting used to. To anyone dealing with math in any way (including programmers) logarithms are absolutely required basic knowledge, don't try to avoid them.

Logarithms were introduced in 1614 by John Napier (white male).

## Details

Different kinds of logarithms exist, distinguished by their base: every logarithm has a base, so we talk about "base 10 logarithm", "base 2 logarithm" etc. (note: base in this context does NOT signify the numeral system base, it's just a name for the number that gets raised to some power). The base is written as a subscript; here we'll write base N logarithm simply as logN (log10, log2 etc.). You may see logarithm written without the base, which either means some implicit base is assumed (often 10 or e, see below) or that the base doesn't matter.

Logarithm is thus not a single function but rather a class of functions. Alternatively you MAY imagine logarithm as a single function of two arguments: the input number x and the base. However conventionally we prefer to view logarithms with different bases as different functions of one argument, perhaps because it rarely happens we need the base to be variable (quite often the base doesn't even matter and it's only a convention of choosing one), and fewer variables means more simplicity, which is always a positive.

Now finally for the proper definition: base N logarithm of number x gives number y, which we write as

logN(x) = y

so that the following holds:

N^y = x

NOTE: Please don't confuse the function N^x with x^N (variable in exponent vs variable in base); inverse function of the former is logarithm N whereas the inverse of the latter is Nth root. These are similar and even the graphs of logarithm and Nth root look similar, but they aren't the same.

So answering the question "What's the base N logarithm of number x?" means answering "N to WHAT power gives x?". Of course we know that both N and y don't have to be just integers, but can be any real number (including fractions, negative numbers etc., however excluding both being zero!), so logarithm is a continuous function, but we also know that (in the realm or real numbers) the operation of raising anything to a power can't ever yield a negative number (and neither zero, unless the base itself was zero), so logarithm is only defined for positive numbers.

NOTE: there exist generalizations such as complex logarithms and it's also possible to have logarithms with bases smaller than 1, but for the sake of simplicity we'll now assume only real number logarithms with a base greater than 1 (which, however, may still be a non-integer).

A small graph will make the best demonstration:

        ^ y
      4 +
        |
      3 +
        |                  ___...''' log2(x)
      2 +            __--''
        |        _.-'
      1 +      .'              _____ log10(x)
        |     / ...----''''''''
    ----+---|'''|---|---|---|---|--> x
       0|  :1   2   3   4
        | ::
     -1 + :
        | :

Here we can see log2(x) and log10(x) plotted, things to observe are mainly these:

- Graph of function logN(x) is a graph of function N^x flipped by the 45 degree axis because logarithm is an inverse of that function (it switches the x and y axes).
- Both go through the point [1,0]. All logarithms give value 0 for input 1 because any number to 0 gives 1.
- Logarithm with base N gives 1 for input N because N^1 = N.
- Similarly both turn from negative to positive y after the point x = 1. Again this is logical: to get values smaller than 1 we have to raise N to a negative power (which makes it a reciprocal fraction).
- Both get closer and closer to minus infinity as they're approaching x = 0 from the right. This represents the fact that raising 1/N to higher and higher powers gets us closer and closer to zero, but we never get there.
- The functions are always increasing, but increase ever more slowly.

By bases some important logarithms are:

- log2 (base 2): Important to programmers (since computers work in base 2), log2(x) for example says how many bits we need to represent x different values, or how many times we can split number x in halves, which is important for computing time complexity.
- log10 (base 10), so called common logarithm: Important because in real life we often use base 10.
- ln (base e), so called natural logarithm: How some nice mathematical properties (for example nice derivative). This is often the default base for a logarithm (i.e. if you see just log, it's usually implied e is the base).

And here is a small table of some logarithm values to further aid making a picture of it all (ln(x) is natural logarithm):

x | ln(x) | log2(x) | log10(x)
--- | --- | --- | ---
0 | ? | ? | ?
0.5 | -0.693147 | -1.000000 | -0.301030
1 | 0.000000 | 0.000000 | 0.000000
1.5 | 0.405465 | 0.584963 | 0.176091
2 | 0.693147 | 1.000000 | 0.301030
2.5 | 0.916291 | 1.321928 | 0.397940
3 | 1.098612 | 1.584963 | 0.477121
3.5 | 1.252763 | 1.807355 | 0.544068
4 | 1.386294 | 2.000000 | 0.602060
4.5 | 1.504077 | 2.169925 | 0.653213
5 | 1.609438 | 2.321928 | 0.698970
5.5 | 1.704748 | 2.459432 | 0.740363
6 | 1.791759 | 2.584963 | 0.778151
6.5 | 1.871802 | 2.700440 | 0.812913
7 | 1.945910 | 2.807355 | 0.845098
7.5 | 2.014903 | 2.906891 | 0.875061
8 | 2.079442 | 3.000000 | 0.903090
8.5 | 2.140066 | 3.087463 | 0.929419
9 | 2.197225 | 3.169925 | 0.954243
9.5 | 2.251292 | 3.247928 | 0.977724
10 | 2.302585 | 3.321928 | 1.000000

One of the most important properties of logarithms, which you absolutely MUST burn into your brain right now, is that logarithm of a product equals sum of logarithms and logarithm of a quotient equals difference of logarithms, i.e.:

logN(a b) = logN(a) + logN(b)*

and

logN(a / b) = logN(a) - logN(b)

Why is this awesome? Well, because now if we can somehow quickly compute logarithm of a number -- for example with the help of precomputed tables, or even just approximately reading it off of the function plot -- we can very quickly multiply or divide numbers simply by looking up the logarithm of each and then adding them or subtracting them (which is not as time consuming as doing multiplication or division) and then mapping the logarithm of the result back. This can be used both by computers and humans to optimize the speed of calculations; in fact this is how the sliding rule works. This way raising a number to a power or finding its root can also be simplified to multiplication or division. (NOTE: someone smart could say we might as well have a precomputed table for multiplication, but that would be a much bigger table due to the fact that we'd need a value for any TWO input numbers, logarithm has just one).

Other important formulas with logarithms include these:

- logN(a^b) = b logN(a)* (simplifies exponentiation to multiplication)
- logA(b) logB(a) = 1*
- logB(x) = logA(x) 1/logA(B)* (allows transforming logarithms between bases)
- logN(N) = 1
- logN(1) = 0
- N^logN(x) = x
- logN(N^x) = x
- ...

To get back to the logarithmic scales for a moment: these are scales whose value at each step increases not by a constant added number, but by multiplying the value of the previous step by some fixed fraction. In graphs such scale may be used on the x or y axis or both, depending on the need -- imagine for instance we were about to plot some exponentially increasing phenomenon, i.e. something that over each period of time (such as a year) grows by some fixed PERCENTAGE (fraction). Example may be the Moore's law stating that the number of transistors in integrated circuits doubles every two years. Plotting this with linear scales we'll see a curve that very quickly shoots up, turning steeper and steeper, creating a very inconvenient, hard to read graph. If instead we used logarithmic scale on the y axis (number of transistors), we'd get a nice straight line! This is because now as we're moving by years on the x axis, we are jumping by orders of magnitude on the y axis, and since that is logarithmic, a jump by order of magnitude shift us a constant step up. This is therefore very useful for handling phenomena that "up closer" need higher resolution and "further away" rather need more more space and bigger "zoom out" on detriment of resolution, such as the map of our Universe perhaps.

        |     | 2^x     /               |            / 2^x
      5 +     |       _/  x          16 +          _/
        |     |      /                  |         /
      4 +     |    _/                 8 +       _/             __
        |    /    /                     |      /         _--'''
      3 +    |  _/         _--'''     4 +    _/     _--''     x
        |   /  /      _--'' log2(x)     |   /   _-''        _____
      2 +  | _/   _-''                2 + _/ _-'   __---''''
        | / /  _-'                      |/ _'  _--'       log2(x)
      1 +'_/ _'                       1 + /  _/
        |/  /                           :   '
      --+--+--+--+--+--+--+--+--+-       ..+--+--+--+--+--+--+--+-
      0 |  1  2  3  4  5  6  7  8          1  2  3  4  5  6  7  8

        | 2^x |  |         /            |        | 2^x  /
      5 +     |  | x     _/          16 +        |    _/  x
        |     |  |      /               |       /    /
      4 +    /   |    _/ log2(x)      8 +      |   _/
        |    |  /    /                  |     /   /           ___
      3 +    |  |  _/                 4 +    |  _/    __---'''
        |   /  /  /                     |  _/  /   _.'    log2(x)
      2 +  |  | _/                    2 + /  _/ _.'
        | /  / /                        |   /  /
      1 +' .'_/                       1 + _/ _/
        : ' /                           :/  /
         ..+--+--+--+--+--+--+--+-       ..+--+--+--+--+--+--+--+-
           1  2  4  8 16 32 64 128         1  2  4  8 16 32 64 128

Graphs of functions 2^x, x and log2(x) with various combinations of linear and logarithmic (base 2) scales on x and y axes: linear, logarithmic y, logarithmic x and both logarithmic (log-log). Notice how one of the curves is always "straightened".

## Programming And Approximations

It won't come as a surprise that we'll find the logarithm function built in most of the popular programming_languages, most often present as part of the standard math library/module. Make sure to check which base it uses etc. C for example has the functions log(x) (natural logarithm), log10(x) and log2(x) under math.h -- if you need logarithm with different base, the simple formula given somewhere above will serve you to convert between arbitrary bases (also shown in an example below).

Should you decide for whatever bold reason to implement your own logarithm, consider first your requirements. If integer logarithm suffices, the straightforward "brute force" way of searching for the correct result in a for loop is quite usable since the number of iterations can't get too high (as by repeated exponentiation we quickly cover the whole range of even 64 bit integers). In C this may be done as follows (we have to watch out for overflows that could get us stuck in an infinite loop; this could also be addressed by using division instead of multiplication, but division can be very slow):

    unsigned int logIntN(unsigned int base, unsigned int x)
    {
      unsigned int r = 0, n = base, nPrev = 0; // nPrev to detect overflow

      while ((n <= x) & (n > nPrev))
      {
        nPrev = n;
        n *= base;
        r++;
      }

      return r;
    }

If we don't insist on having the base a variable it's better to have it constant, the function will most likely get faster (passing one fewer argument, compiler can optimize expression with the constant etc.) -- especially log2 can be optimized by using a bit shift and being able to simplify everything like this:

    unsigned int logInt2(unsigned int x)
    {
      unsigned int r = 0;

      while (x > 1)
      {
        x >>= 1;
        r++;
      }

      return r;
    }

As always, look up tables may help create extremely fast versions for the price of some memory.

Mainstream way of implementing floating point logarithm is probably through Taylor series and similar infinite series such as (for natural logarithm):

ln(x) = 2 ((x-1)/(x+1) + 1/3 * ((x-1)/(x+1))^3 + 1/5 * ((x-1)/(x+1))^5 + ...)*

Here this formula is used to implement a somewhat workable floating point natural logarithm and general base logarithm (a bit dirty due to hardcoded constants, but can be a start):

    double logFloatE(double x)
    {
      double r = 0;

      if (x > 2) // for larger values precision decreases
        return logFloatE(x * 3.0/4.0) - logFloatE(3.0/4.0);

      x = (x - 1) / (x + 1);

      for (double i = 1; i < 8; i += 2) // 8 arbitrarily chosen
      {
        r += x / i;
        x *= x * x;
      }

      return 2 * r;
    }

    double logFloatN(double base, double x)
    {
      return logFloatE(x) / logFloatE(base);
    }

As for approximations: unfortunately good ones are often plagued by narrow interval of convergence. Attempts at construction of a function resembling logarithm may perhaps start with a similarly shaped function 1 - 1/x, then continue by pimping it up and adding correcting expressions until it looks cool. This may lead for example to the following expression: { Made by me. ~drummyfish }

log10(x) ~= 3.0478 + 0.00001 x - 205.9 / (x + 100) - (1233 * x + 10) / (625 * (x + 1) * x)*

It's not very precise but the advantage is that it looks reasonable on a wide interval from 0 up to many thousands: before x gets to higher hundreds the error is somewhere around 3%, then around 2000 gets to some 10% and around 10000 to approx 20% where it then seems to stay for a very long time.

Should you have the pow (or even just exp) function at hand (which can itself be approximated), you could likely use it to implement floating point logarithm also through binary search with delta, Newton's method or something similar.

## See Also

- power law

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