The same value can also be represented in normalized form with 1.2345 as the fractional coefficient, and +2 as the exponent (and 10 as the base):

The confusion is because scientific notation and floating-point representation are log-linear, not logarithmic. To multiply two numbers, given their logarithms, one just adds the characteristic (integer part) and the mantissa (fractional part). By contrast, to multiply two floating-point numbers, one adds the exponent (which is logarithmic) and multiplies the significand (which is linear).