How are the logs coming about? thanks

Question

Sadie Alexander · Accepted Answer

A few thoughts...

I am not exactly sure what you are asking, but here's something of what logarithms are about... Let #b > 1# be a constant. Consider the function: #f(x) = b^x# This is a continuous, one to one, strictly monotonically increasing function from #(-oo, oo)# onto #(0, oo)# looking something like this: graph{2^x [-9.58, 10.42, -2.4, 7.6]} It has the interesting property that: #f(x+y) = b^(x+y) = b^x b^y = f(x)f(y)# for any #x, y in (-oo, oo)# The inverse of #f(x)# has a graph like this: graph{2^y-x = 0 [-4.08, 15.92, -4.12, 5.88]} which is a continuous, one to one, strictly monotonically increasing function from #(0, oo)# onto #(-oo, oo)# Given any #u, v in (0, oo)#, let: #x = f^(-1)(u)# #y = f^(-1)(v)# Then we find: #f^(-1)(uv) = f^(-1)(f(x)f(y)) = f^(-1)(f(x+y)) = x+y = f^(-1)(u)+f^(-1)(v)# This inverse function #f^(-1)(x)# is the logarithm base #b# and this property we have found is: #log_b(uv) = log_b(u)+log_b(v)# a fundamentally useful property of logarithms. This property of logarithms was discovered by John Napier in the 17th century and used in the form of slide rules and tables to help perform calculations.