The key that TFA probably missed is that the bit pattern following the exponent (which is by definition an approximation of the logarithm) is the "linear" bit o

The key that TFA probably missed is that the bit pattern following the exponent (which is by definition an approximation of the logarithm) is the "linear" bit of the "piecewise linear approximation", while the exponent is the "piecewise" bit. That's really all the answer needed to explain in the end. "Why is an IEEE 754 positive [normal] number interpreted as an integer a piecewise linear approximation of its logarithm?" -> "Because the bit pattern for it will have an exponent followed by a mantissa, where the exponent by definition is a very rough approximation of the logarithm, and where the mantissa can be interpreted as a linear interpolation between the exponent and the next exponent, thus the two pieces as the "piecewise" (exponent) and "linear" (mantissa) approximation of the number's logarithm." The only remaining bits of interest is the bias of the exponent and the assumed-1 leading digit of the mantissa being left out. The latter is a compression technique (why wastefully include a bit that's always set or always reset?) that happens to help here (avoiding having to shift the mantissa bits by one). The former is easy enough to deal with.