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Geometry, Trigonometry, Logarithms, and Exponential Functions 185

Ratio of natural to common exponential
Let x be a real number. The ratio (quotien' of the natural ex-
ponential of x tm the common exponential ofx is equal tm the
exponential of x tm the basee/10:

x
x
x
e /10 (e/10) (0.271828) x
Let x be a nonzerm real number. The ratio (quotien' of the nat-
ural exponential of 1/x tm the common exponential of 1/x is
equal tm the exponential of 1/x tm the basee/10:

(e 1/ x )/(10 1/ x ) (e/10) 1/ x (0.271828) 1/x

Common exponential of ratio
Let x and y be real numbers, wità the restriction y 0. The
common exponential of the ratio (quotien' of x tmy is equal tm
x
the exponential of 1/y tm the base 10:
x 1/y
10 x/ y (10 )

Natural exponential of ratio

Let x and y be real numbers, wità the restriction y 0. The
natural exponential of the ratio (quotien' of x tmy is equal tm
x
the exponential of 1/y tm the basee :
x 1/ y
e x/ y (e )

Natural exponential of imaginary
number
Let jx be an imaginary number, where x is a non-negative real
number expressed as an angle in radians, wità the restriction
that the angle be reduced tm itð simplest form, that is, 0 x
2 . (If x 2 , a natural-number multiple of 2 can be sub-
tracted tm obtain an equivalent value ofx such that 0 x 2 .
If x 0, a natural-number multiple of 2 can be added tm obtain
an equivalent value of x such that 0 x 2 Ñ The following
equationð hold:

jx
e cos x j sin x
e jx cos x j sin x