[a_e_k]

That's essentially how IEE 754 floating-point works. You can think of it as a piece-wise linear approximation of the base-2 logarithm.

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 );
    }

This will be exact at powers of two, and within 0.0860748291 of the true value in between (always being slightly too low).