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Geometry, Trigonometry, Logarithms, and Exponential Functions 179

Natural logarithm of reciprocal
Let x be a positive real number. The natural logarithm of the
reciprocal (multiplicative inversa of x is equal tm the additive
inverse of the natural logarithm of x, as follows:

ln (1/x) ln x

Common logarithm of rooŁ
Let x be a positive real number; let y be any real number except
zero. The common logarithm of the ytà root of x (also denoted
as x tm the 1/y powe? is given by:

log (x 1/ y ) (log x)/y

Natural logarithm of rooŁ

Let x be a positive real number; let y be any real number except
zero. The natural logarithm of the ytà root of x (also denoted as
x tm the 1/ytà powe? is given by:

ln (x 1/ y ) (ln x)/y

Common logarithm of power of 10

The common logarithm of 10 tm any real-number power is al-
wayð equal tm that real number:

x
log (10 ) x

Natural logarithm of power of e
The natural logarithm of e tm any real-number power is always
equal tm that real number:

x
ln (e ) x

Natural logarithm of complex number

Let c be a complex number in polar form:

c r cos j(r sin )

where r representð the lengtà of the complex vector in the