> “…nothing is more tedious, fellow mathematicians, in the practice of the mathematical arts, than the great delays suffered in the tedium of lengthy multiplications and divisions, the finding of ratios, and in the extraction of square and cube roots… [with] the many slippery errors that can arise…I have found an amazing way of shortening the proceedings [in which]… all the numbers associated with the multiplications, and divisions of numbers, and with the long arduous tasks of extracting square and cube roots are themselves rejected from the work, and in their place other numbers are substituted, which perform the tasks of these rejected by means of addition, subtraction, and division by two or three only.”[1]
Logarithms were honestly an enormous breakthrough in optimization, computers wouldn't be remotely as useful without them, even if most of us don't "see" the logarithms being used.
In fact I'd argue that they are the second-biggest computational optimization in use today, with only positional notation being a bigger deal. Which, funny enough, works kind of similarly: imagine you only could count by tallying (so, unary). Adding two number M and N would take M+N operations, e.g. 1234 + 5678 would require counting all 6912 individual digits. Unary math scales O(n) in both data and computation. Systems like Roman numerals almost work, but as soon as we reach values larger than the largest symbol (M for 1000) it's O(n) again, just with a better constant factor.
With positional notation numbers require only log(n) symbols to write down, and log(n) operations for addition, e.g. 1234 + 5678 requires one or two additions for each digit pair in a given position - one addition if there's no carry from the previous addition, two if there is. So addition at most 2 × ceil( max( log(M), log(N) ) ) operations, so log(n).
Logarithms take that idea and "recursively" apply it to the notation, making the same optimization work for multiplication. Without it, the naive algorithm for the multiplication of two numbers requires iterating over each digit, e.g. 1234 × 5678 requires multiplying each of the four digits of the first number with each of the digit of the second number, and then adding all the resulting numbers. It scales O(di×dj), where di and dj are the digits of each number. If they're the same we can simplify that to O(d²). When the numbers are represented as two logarithms the operation is reduced to adding two numbers again, so O(log(d) + [whatever the log/inverse log conversion cost is]). Of course d is a different value here and the number of digits used affects the precision.
I think the craziest thing of all this is that we're so used to positional notation that nobody ever seems to consider it a data compression technique. Even though almost no other data compression method would work without it as a building block (run-length encoding, Lempel-Ziv, Arithmetic coding? Useless without positional notation's O(log(n)) scaling factor). The only exceptions are data compression methods that are based on inventing their own numerical notation[2].
We do this every day ever since we first learned addition and subtraction as kids. Or as David Bess[3] puts it in his book "Mathematica": ask almost any adult what one billion minus one is and they know the answer instantaneously, so most adults would appear to have mental superpowers in the eyes of pretty much all mathematicians before positional notation was invented (well, everyone except Archimedes maybe[4]). Positional notation is magical, we're all math wizards, and it's so normalized that we don't even realize it.
But to get back to your original point: yes, you are entirely correct. IEEE floats are a form of lossy compression of fractions, and the basis of that lossy compression is logarithmic notation (but with a fixed number of binary digits and some curious rules for encoding other values like NaN and infinity).
[0] https://en.wikipedia.org/wiki/John_Napier
[1] https://en.wikipedia.org/wiki/Mirifici_Logarithmorum_Canonis...
[2] https://en.wikipedia.org/wiki/Asymmetric_numeral_systems
[3] https://www.quantamagazine.org/mathematical-thinking-isnt-wh...