Logarithm Function (Python, functional)

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Other implementations: Pascal | Python | Python, functional

Calculate base-10 logarithms to a specified number of decimal places.

Contents 1 theory 2 practice 3 wrapping up 4 agendum

theory

If one wishes to use logarithms in Python, then the built-in math.log10 is the obvious solution.

Here we take advantage of the algebraic structure of to produce an entirely functional expression approximating log10 to a given number of digits.

practice

For much of this program, we deal with pairs (x,l), where l accumulates an integer approximation to the logarithm, and x encodes the remainder of the argument for which we seek the logarithm.

• shift, in expressing the identity l + log₁₀x = l + s + log₁₀x / 10^s, shifts the decimal point of x.
• scale is a shift with an argument of -1, 0, or 1, either bringing the x closer to units magnitude while incrementing (or decrementing) l, or, with an s at 0, returning x and l unchanged.
• ipart is the value which iterated scales converge to, at which point l is the integer part of the logarithm.

<<extracting the integer part>>=
shift = lambda x,l,s: (x/(10.0**s), l+s)
scale = lambda x,l: shift(x, l, (x >= 10)-(x < 1))
ipart = lambda x: fix(scale, (x,0))

To extract the fractional digits, we repeat the steps above for each fractional decimal place. A single digit is l, and the digits which come after it are handled by shifting the effective decimal place in the recursion: logx¹⁰ = 10 * logx

In a lazy language, the restriction d on number of decimal places extracted could be handled in the top-level expression, but as python is eager, we must propagate it through the entire expression, truncating with a value of 0.0 after the last decimal place desired.

<<approximating log10>>=
digit = lambda x,l,d: l + alog(x**10, d-1)/10.0
alog  = lambda x,d=12: (d>=0) and digit(*ipart(x)+(d,)) or 0.0

Question: why not write (d<0) and 0.0 or digit(*ipart(x)+(d,))?

Question: which instances of 10 in the code above refer to the base of the logarithm, and which refer to the decimal output?

wrapping up

We now provide the ancillary detail of arranging for a computation to converge.

Exercise: in this case, there is no danger of scale entering a loop. Write a more general fixpoint, which detects convergence to a loop and terminates properly. In such cases, does it matter which loop element is returned?

<<converging to a fixpoint>>=
fix   = lambda f,v1,v0=None: (v0 == v1) and v0 or fix(f, f(*v1), v1)

Finally, we wrap everything up in a module, which, when run, uses the same test vectors as Logarithm Function (Python)

<<alog10.py>>=
converging to a fixpoint
extracting the integer part
approximating log10
if __name__ == "__main__":
    value = 4.5
    print alog(value, 3)
    print alog(value, 6)
    print alog(value)

... and add a quick sanity check:

<<alog10.py>>=
    e = 2.7182818284590451
    print alog(10**e)
    print 10**alog(e)
    print e

The output should be:

0.653
0.653212
0.653212513775
2.71828182846
2.71828182846
2.71828182846

agendum

• Rewrite this article according to the following exercise, but not until several years after the appropriate python version has been available.

Exercise: rewrite alog and fix to use the ternary operator.

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