The natural logarithm of a positive real number a is denoted by ln a and defined byso it is the (signed) area enclosed by the curve y=1/x, the x-axis, and the lines x=1 and x=a. If a>1 then ln a>0, and if a<1 then ln a<0. Also ln 1=0. The natural logarithm is related to the exponential function by exp(ln x)=x for all x>0 and ln{exp(x)}=x for all values of x. It is also referred to as the logarithm to base e. For historical reasons a natural logarithm is sometimes referred to as a Napierian logarithm, after the Scottish mathematician John Napier (1550–1617).