From Roots and rational powers of positive numbers
to real powers of positive numbers.
-
ln(a m) = m ln (a) implies a m
= exp( ln(a m)) = exp(m ln (a)). Eg (1.7)3
= exp( 3 ln(1.7)).
-
Now b = a 1/m when and only when b m
= a. The latter implies ln(a) = ln(b m) = m ln (b) and hence ln(b)
= (1/m) ln (a). So b = exp (ln(b)) = exp( (1/m) ln(a) ) = a
1/m
Eg. 8 1/3 = exp( (1/3) ln(8)
Now if m and n have no common divisors, and n is nonzero, let the
m/n power of a m/n = (a m)1/n
Then a m/n = (a m)1/n = exp(
(1/n) ln(a m)) = exp( (1/n)m ln(a m)) = exp( (m/n) ln(a))
Roots and rational powers of positive numbers
How to compute using logs and exponentials
-
a m = exp( ln(a m))
= exp(m ln (a))
-
a 1/m = exp( (1/m) ln(a) )
-
a m/n = (a m)1/n
= exp( (m/n) ln(a))
|
EG: 8 1/3 = exp( (1/3) ln(8)
EG 8 2/3 = exp( (2/3) ln(8)) = ( 8 1/3)
2
Exponentials of Real Numbers a x = exp( x ln(a))
For x = m/n and a > 0, a x = a m/n
= exp( (m/n) ln(a)) = exp( x ln(a)). This suggests putting a
x = exp( x ln(a)) for x irrational. Then
a x = exp( x ln(a)) for all real x for a
> 0
and not only for rational numbers. From this definition,
ln a x = x ln(a). Therefore loga(a x)
= x because loga(x) = ln(x)/ln(a).
Properties of Exponentials
Now (a x)y = exp(y ln(a x
)) = exp(y x ln(a )) = a yx = a xy
Therefore
(a x)y = a
xy (Exponential of an exponential)
Now a xay = exp(x
ln(a)) · exp(y ln(a) = exp(x ln(a)+y ln(a)) = exp( (x +y )ln(a) ) = a x+y Therefore
has the exponential property
a xay = a x+y
for all real numbers x and y when a > 0.
Now for the natural number e = exp(1) =
2.718281828... (irrational, deci), the natural logarithm of e, ln
(e) = 1 Therefore
e x = exp( x) for all real x when a
> 0
as a x = exp( x ln(a)). Calculators often have a
button marked e x for the evaluation of the exponential
function exp( x)
Caution: the capital EXP on some calculators will not
help you with the calculation of exp(x). Use the button marked e
x instead.
|