Natural Logarithm – ln(x)

Natural logarithm is the logarithm to the base e of a number.

Definition of natural logarithm

When

e ^y = x

Then base e logarithm of x is

ln(x) = log_e(x) = y

The e constant or Euler’s number is:

e ≈ 2.71828183

Ln as inverse function of exponential function

The natural logarithm function ln(x) is the inverse function of the exponential function e^x.

For x>0,

f (f ^-1(x)) = e^ln(x) = x

Or

f ^-1(f (x)) = ln(e^x) = x

Natural logarithm rules and properties

Rule name	Rule	Example
Product rule	ln(x ∙ y) = ln(x) + ln(y)	ln(4 ∙ 5) = ln(4) + ln(5)
Quotient rule	ln(x / y) = ln(x) - ln(y)	ln(4 / 5) = ln(4) - ln(5)
Power rule	ln(x y) = y ∙ ln(x)	ln(37) = 7 ∙ ln(3)
ln derivative	f (x) = ln(x) ⇒ f ' (x) = 1 / x
ln integral	∫ ln(x)dx = x ∙ (ln(x) - 1) + C
ln of negative number	ln(x) is undefined when x ≤ 0
ln of zero	ln(0) is undefined
ln of one	ln(1) = 0
ln of infinity	lim ln(x) = ∞ ,when x→∞
Euler's identity	ln(-1) = iπ

Logarithm product rule

The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.

log_b(x ∙ y) = log_b(x) + log_b(y)

For example:

log₈(4 ∙ 5) = log₈(4) + log₈(5)

Logarithm quotient rule

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.

log_b(x / y) = log_b(x) – log_b(y)

For example:

log₈(4 / 5) = log₈(4) – log₈(5)

Logarithm power rule

The logarithm of x raised to the power of y is y times the logarithm of x.

log_b(x ^y) = y ∙ log_b(x)

For example:

log₈(3⁹) = 9 ∙ log₈(3)

Derivative of natural logarithm

The derivative of the natural logarithm function is the reciprocal function.

When

f (x) = ln(x)

The derivative of f(x) is:

f ‘ (x) = 1 / x

Integral of natural logarithm

The integral of the natural logarithm function is given by:

When

f (x) = ln(x)

The integral of f(x) is:

∫ f (x)dx = ∫ ln(x)dx = x ∙ (ln(x) – 1) + C

Ln of 0

The natural logarithm of zero is undefined:

ln(0) is undefined

The limit near 0 of the natural logarithm of x, when x approaches zero, is minus infinity:

Ln of 1

The natural logarithm of one is zero:

ln(1) = 0

Ln of infinity

The limit of natural logarithm of infinity, when x approaches infinity is equal to infinity:

lim ln(x) = ∞, when x→∞

Complex logarithm

For complex number z:

z = re^iθ = x + iy

The complex logarithm will be (n = …-2,-1,0,1,2,…):

log z = ln(r) + i(θ+2nπ) = ln(√(x²+y²)) + i·arctan(y/x))

Graph of ln(x)

ln(x) is not defined for real non positive values of x:

Natural logarithms table