Suppose I want to multiply 8 and 256. I note that 8 × 256 = 2

^{3}× 2^{8}= 2^{11}= 2048. I have turned multiplication into addition. A voice in my head says, “Hey, that could be a useful trick.” Another voice says “What if you want to multiply 7 and 652? Stupid!” It is true that 7 = b^{x}or 652 = b^{y }have no whole number solutions (except the obvious and useless b = b^{1}.) But we do not have to restrict ourselves to whole numbers.
Consider the equations √10 × √10 = 10 and 10

^{0.5}× 10^{0.5}= 10^{1}= 10. In these equations √10 and 10^{0.5}have exactly the same job. We can consider them equivalent. Hence the exponent ½ (or 0.5) represents a square root. Similarly the exponent ⅓ represents a cube root. In general, the exponent n/m represents the m^{th}root of a number to the power n. For example, 10^{5/6}is the sixth root of 100 000. Note that 100 000 = 10^{5}.
Lets look again at the product of 7 and 652.

P = 7 × 652

P = 10

^{a}× 10^{b}= 10^{a+b}.
I want 7 = 10

^{a}and I want 652 = 10^{b}. In other words I want the logarithms of 7 and of 652 in base 10. By definition of a logarithm y = b^{x}ó log_{b}y = x. Therefore 7 = 10^{a}ó log_{10}7= a and 652 = 10^{b}ó log_{10}652= b. The numbers a and b are 0.845 098 040 and 2.814 247 595 respectively.
Therefore P = 10

^{ 0.845 098 040 +2.814 247 595 }= 10^{3.659345635}= 4563.999 = 4564
Ok, that’s impressive. But where did you get log

_{10}7= 0.845 098 040 and log_{10}652 = 2.814 247 595? Agreed , that’s the hard part. John Napier was the first person to calculate a table of logarithms. He published his table in the year 1614; after 20 years of work!
Picture from https://math.stackexchange.com/questions/47927/motivation-for-napiers-logarithms

Napier's "invention was quickly and widely met with acclaim. The works of Bonaventura
Cavalieri (Italy), Edmund Wingate (France), Xue Fengzuo (China), and Johannes
Kepler's

*Chilias logarithmorum*(Germany) helped spread the concept.” https://en.wikipedia.org/wiki/History_of_logarithms
Napier’s original table did not use base 10. A few years later Napier worked with Henry Briggs
to produce a table of common (base 10) logarithms.
Briggs continued after Napier died and published
their table in 1624.

Picture from

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