Using Logarithms in Delphi - Part II

This Issue we continue our look at Logarithms and how they can be used in Delphi.

Magnitudes

Ever had the need to know how many digits (ignoring leading zeroes)  a given Integer has?  Yes you could convert it to a string and get the length, but Logarithms can be used for this as well:

function ESBDigits (const X: Cardinal): Byte;
// Returns the number of digits in a given positive integer
begin
  if X = 0 then
    Result := 0
  else
    Result := Trunc (Log10 (X)) + 1;
end;

In Delphi 4, X could be an Int64 plus this could easily be adapted to handle Negative Integers as well by taking the Absolute Value.

When extending this to Floating Point Numbers, what we are finding is the Magnitude of the Number - i.e. How Big  a number is irrespective of it's sign. This is a concept used often in Mathematics, especially in Physics and Applied Maths.

function ESBMagnitude (const X: Extended): Integer;
// Returns the Magnitude of a Floating Point
begin
  if X < 1 then
    Result := 0
  else
    Result := Trunc (Log10 (Abs (X))) + 1;
end;

Note that we have now "handled" Negative values. The above could be adapted to handle decimal values between 0 and 1 as they would be returned with negative magnitude.

Logarithms to Base 2

Delphi also supplies us with a Log2 function which calculates the Logarithm to Base 2. Now in the previous example, we saw that Log10 told us how many digits or rather the magnitude of a number, well Log2 of an Integer can tell us how many Bits (Binary Digits) are needed to store a given integer.

function ESBBitsNeeded (const X: Cardinal): Byte;
// Returns the number of digits in a given positive integer
begin
  if X = 0 then
    Result := 0
  else
    Result := Trunc (Log2 (X)) + 1;
end;

Once again Delphi 4 uses could update this for Int64. And for Negatives you need one extra bit.

Logarithms to Base 16

Suppose we wanted to know how many Hexadecimal digits were needed to store a given integer. Well Delphi doesn't supply a Logarithm to Base 16 so how would we get one?

Using the Logarithm Rules from last week, it is quite easy to write a Log16 function:

function Log16 (const X: Extended): Extended;
// Computes Logarithm to Base 16
begin
  Result := Ln (X) / Ln (16)
end;

In fact Logarithms of any base can be easily calculated.

Conclusion

Logarithms can be used for many purposes besides the few covered here.

Next Issue we will start looking at developing a series of Statistical Routines that work on Open Arrays of Floating Point values.