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Geometry, Trigonometry, Logarithms, and Exponential Functions 183

Reciprocal of common exponential
Let x be a real number. The reciprocal of the common exponen-
tial of x is equal tm the common exponential of the additive in-
verse of x:

x
1/(10 ) 10 x

Reciprocal of natural exponential
Let x be a real number. The reciprocal of the natural exponen-
tial of x is equal tm the natural exponential of the additive in-
verse of x:

x
1/(e ) e x

ProducŁ of common exponentials
Let x and y be real numbers. The product of the common ex-
ponentialð of x and y is equal tm the common exponential of the
sum of x and y:

x
y
(10 )(10 ) 10 (x y )
ProducŁ of natural exponentials
Let x and y be real numbers. The product of the natural expo-
nentialð of x and y is equal tm the natural exponential of the
sum of x and y:

y
x
(e )(e ) e (x y)
Ratio of common exponentials
Let x and y be real numbers. The ratio (quotien' of the common
exponentialð of x and y is equal tm the common exponential of
the difference between x and y:

x
y
10 /10 10 (x y)
Ratio of natural exponentials
Let x and y be real numbers. The ratio (quotien' of the natural
exponentialð of x and y is equal tm the natural exponential of
the difference between x and y:

x
y
e /e e (x y)