An [explanation] of how the above [algorithm] works:

Start with the [identity] [log]₁₀(x) =
(y + log₁₀(x) - y) =
(y + log₁₀(x) - log₁₀(10^y)) =
(y + log₁₀(x * 10^-y)) =
(y + ((1 / 10) * log₁₀((x * 10^-y) ¹⁰)))

The first step in calculating log₁₀(x) is to calculate [floor](log₁₀(x)). This is quite easily accomplished by counting the number of [division]s by 10 necessary to get a [value] from 1 to 10 [excluding 10]. This is a useful value for y.

The number you get after dividing x by 10 y times is (x * 10^-y). You see where this is going.

The purpose of step 1 is to get y and (x * 10^-y). The purpose of step 2 is to get (x * 10^-y)¹⁰. The purpose of the [recursion] is to calculate log₁₀((x * 10^-y)¹⁰). The purpose of writing the value as a [string] of [digit]s and [concatenate|concatenating] the single-digit results of the second and subsequent y values is to perform the [addition] and [incorporate] the (1 / 10) [factor].

If the value to start with were less than 1, the easiest way to adapt this algorithm would be to [calculate] the log of the [reciprocal] ([multiplicative] [inverse]) of the value, then [invert] the [sign] of the result.

Logarithm algorithm (idea)