Logarithm Function (Python)

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Other implementations: Pascal | Python | Python, functional

Logarithm function

The logarithm function performs a mathematical operation that is the inverse of an exponentiation function (raising a constant, the base, to a power). The logarithm of a number x in base b is any number n such that x = bⁿ. It is usually written as

Finding the logarithm in base b of a positive real number x can be done in two parts:

Finding the integer part of the logarithm
Finding the fractional part of the logarithm

1. Finding the integer part

Finding the integer part can be done by simply dividing the number x by the base b until it is no longer larger than the base b. The integer part is the number of iterations we do the division.

For example: Find log base 3 of 161.64


iteration no 1		$161.64 / 3 = 53.88$
iteration no 2		$53.88 / 3 = 17.96$
iteration no 3		$17.96 / 3 = 5.9866$
iteration no 4		$5.9866 / 3 = 1.9955$

As it takes 4 iterations, the integer part of the logarithm is 4.

2. Finding the fractional part

To find the fractional part, we use the fact that multiplying a logarithm number by two is equivalent to squaring.

Example: find fractional part of 161.64 in log base 3

thus

Next we square both sides

Because 3.98 is greater than 3, we can write:

thus

Next we concentrate on $f 1$

becomes

Next we repeat the trick of squaring both sides

Because 1.76 is less than 3, we can write:

thus:

Next we concentrate on $f 2$

becomes

With the general procedure established above, we can calculate , , , and so on.

Recapping, we have:

and

thus creating

In general, we have

Where is either 1 or 0

Source Code

<<logN.py>>=
#!/usr/bin/python
from __future__ import division
import math
def logN(X, base=math.e, epsilon=1e-12):
  # logN is logarithm function with the default base of e
  integer = 0
  if X < 1 and base < 1:
    raise ValueError, "logarithm cannot compute"
  while X < 1:
    integer -= 1
    X *= base
  while X >= base:
    integer += 1
    X /= base
  partial = 0.5               # partial = 1/2 
  # list = []                   # Prepare an empty list, it seems useless
  X *= X                      # We perform a squaring
  decimal = 0.0
  while partial > epsilon:
    if X >= base:             # If X >= base then a_k is 1 
      decimal += partial      # Insert partial to the front of the list
      X = X / base            # Since a_k is 1, we divide the number by the base
    partial *= 0.5            # partial = partial / 2
    X *= X                    # We perform the squaring again
  return (integer + decimal)
if __name__ == '__main__':
  value = 4.5
  print "       X  = ", value
  print "    ln(X) = ", logN(value)
  print "  log4(X) = ", logN(value, base=4)

Sample Run

$ python logN.py
       X = 4.5
    ln(X)= 1.50407739678
  log4(X)= 1.08496250072

Extension

Given the equation:

The algorithm above computes

However, by modifying the algorithm slightly, it's possible to compute

This will allow us to stop the computation at a particular decimal place.

Source Code

<<logNdp.py>>=
#!/usr/bin/python
from __future__ import division
import math
# Note: compute the log of X>=1. Doesn't work for X<1. Ex: logN(0.5) returns -1.698970... instead of -0.301029...
def logN(X, base=math.e, decimalplaces=12):
  integer = 0
  while X < 1:
    integer -= 1
    X *= base
  while X >= base:
    integer += 1
    X /= base
  result = str(integer) + '.'
  while decimalplaces > 0:
    X = X ** 10                     # Calc X to the 10th power
    digit = 0
    while X >= base:
      digit += 1
      X /= base
    result += str(digit)
    decimalplaces -= 1
  return result
if __name__ == '__main__':
  value = 4.5
  print "                        X  = ", value
  print " 3 decimal places   LOG(X) = ", logN(value, base=10, decimalplaces=3)
  print " 6 decimal places   LOG(X) = ", logN(value, base=10, decimalplaces=6)
  print "12 decimal places   LOG(X) = ", logN(value, base=10)

Sample Run

$ python logNdp.py
                        X  = 4.5
 3 decimal places   LOG(X) = 0.653
 6 decimal places   LOG(X) = 0.653212
12 decimal places   LOG(X) = 0.653212513775

To compute logarithm of 0<X<1 and X>=1

(created in Python 3.1 using modified source as contained on this page)

from __future__ import division
import math
from os import * 
def logN(X, base=math.e, decimalplaces=12):
  if X>=1:
    integer = 0
    while X < 1:
      integer -= 1
      X *= base
    while X >= base:
      integer += 1
      X /= base
    result = str(integer) + '.'
    while decimalplaces > 0:
      X = X ** 10                     # Calc X to the 10th power
      digit = 0
      while X >= base:
        digit += 1
        X /= base
      result += str(digit)
      decimalplaces -= 1
    return result
  #end if
  else:
    reverse=X*base
    reverse=base/reverse
    result = logN(reverse,base,decimalplaces)
    result = float(result) * (-1.0)             #must cast result as float to calculate.
    return result
################
# Main Program #
################
value=1  #sentinel value to start loop
while (value > 0):
    print("Enter the value for log (base 10), enter zero or a negative to quit ->")
    value=float(input())
    base = 10
    if (value > 0):
        print("                        X  = ", value)
        print(" 3 decimal places   LOG(X) = ", logN(value, base=10, decimalplaces=3))
        print(" 6 decimal places   LOG(X) = ", logN(value, base=10, decimalplaces=6))
        print("12 decimal places   LOG(X) = ", logN(value, base=10, decimalplaces=12))
    #end if
#end while
print("Thanks for using this program!")

Example Run

4.5
                        X  =  4.5
 3 decimal places   LOG(X) =  0.653
 6 decimal places   LOG(X) =  0.653212
12 decimal places   LOG(X) =  0.653212513775
Enter the value for log (base 10), enter a negative to quit ->
.5
                        X  =  0.5
 3 decimal places   LOG(X) =  -0.301
 6 decimal places   LOG(X) =  -0.301029
12 decimal places   LOG(X) =  -0.301029995663
Enter the value for log (base 10), enter a negative to quit ->

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Category: Exponential and logarithm functions

Logarithm Function (Python)

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