Classic examples include:

- time domain / frequency domain

- lookup table / functional form

- traditional binary representation / gray codes

The encoding in terms of prime exponents is known as the Godel encoding, and its a fairly important step in proving things like Godel's incompleteness theorems and the undecidability of the Halting problem.

Essentially it's a one-to-one map between natural numbers and finite sequences of natural numbers (since you can encode (a,b,c,d...) as 2^a × 3^b × 5^c × 7^d etc), which turns out to be mathematically a convenient thing to have.

    A × B = C 

    log A + log B = log C

    D^E = F

    E × log D = log F

Relies on you either having stored the needed data or efficiently calculating logirithms and antilogirithims.

https://www.cuemath.com/algebra/log-table/

For non-negative values, the exponent (upper) bits give the integer part of the logarithm and the significand (lower) bits approximate the fractional part (i.e., the mantissa).

In other words, you can do:

    float approxLog2( float value )
    {
        assert( value > 0.0f );
        auto bits = std::bit_cast< int32_t >( value );
        auto exponent = ( bits >> 23 ) - 127;
        auto significand = bits & ( ( 1 << 23 ) - 1 );
        return exponent + significand / static_cast< float >( 1 << 23 );
    }