# The Natural Logarithm is the Limit of the Integrals of Powers

The Natural Logarithm is the Limit of the Integrals of Powers

Assume that and that .

a≠0

x>0

The integral of is +C, where is an arbitrary constant. The integral of is , where again is an arbitrary constant and is the natural logarithm of , often written as .

x

a-1

x

a

a

C

x=

-1

1

x

log(x)+C'

C'

log(x)

x

ln(x)

When is close to zero, and are close, so there must be some connection between their integrals!

a

x

a-1

x

-1

Choose and so that the two integrals are both zero at . The integrals are then - and . For close to zero these functions are very close; in symbols, .

C=-

1

a

C'=0

x=1

x

a

a

1

a

log(x)

a

lim-=log(x)

a0

x

a

a

1

a

Using the difference quotient for the derivative of the base- exponential function with respect to (not ) and using instead of the more usual gives . This is more usually written with as the variable: u=ulog(u), with the special case e=e.

x

f(b)=x

b

b

x

a

h

f'(b)=lim=lim=xlim=xlog(x)

a0

f(b+a)-f(b)

a

a0

x-x

b+a

b

a

b

a0

x-1

a

a

b

x

d

dx

x

x

d

dx

x

x