A Multiplication Shortcut

Suppose I want to multiply 8 and 256.  I note that 8 × 256 = 2³ × 2⁸ = 2¹¹ = 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 = b¹.)  But we do not have to restrict ourselves to whole numbers.

Consider the equations √10 × √10 = 10 and 10^0.5 × 10^0.5 = 10¹ = 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 = 10⁵.

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 log₁₀ 7= a   and 652 = 10^b ó log₁₀ 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 log₁₀ 7= 0.845 098 040  and log₁₀ 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

http://www.cbi.umn.edu/hostedpublications/Tomash/Images%20web%20site/Image%20files/B%20Images/pages/Briggs.Arithmetica%20logarithmica.1624.critica%20logarithms.htm