Natural Logarithm - ln(x)Natural logarithm is the logarithm to the base e of a number.

Natural logarithm (ln) definition Natural logarithm (ln) rules & properties

o Derivative of natural logarithm (ln)o Integral of natural logarithm (ln)

Graph of ln(x) Natural logarithms (ln) table Natural logarithm calculator

Definition of natural logarithmWhen

e y = xThen base e logarithm of x is

ln(x) = loge(x) = yThe e constant or Euler's number is:e ≈ 2.71828183

Ln as inverse function of exponential functionThe natural logarithm function ln(x) is the inverse function of the exponential function ex.For x>0,

f (f -1(x)) = eln(x) = xOr

f -1(f (x)) = ln(ex) = x

Natural logarithm rules and propertiesRule name Rule Example

Product rule ln(x ∙ y) = ln(x) + ln(y) ln(3 ∙ 7) = ln(3) + ln(7)

Quotient rule ln(x / y) = ln(x) - ln(y) ln(3 / 7) = ln(3) - ln(7)

Power rule ln(x y) = y ∙ ln(x) ln(28) = 8∙ ln(2)

ln derivativef (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 zeroln(0) is undefined

ln of one ln(1) = 0ln of infinity lim ln(x) = ∞ ,when x→∞

Logarithm product ruleThe logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.

logb(x ∙ y) = logb(x) + logb(y)For example:

log10(3 ∙ 7) = log10(3) + log10(7)Logarithm quotient ruleThe logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.

logb(x / y) = logb(x) - logb(y)For example:

log10(3 / 7) = log10(3) - log10(7)Logarithm power ruleThe logarithm of x raised to the power of y is y times the logarithm of x.

logb(x y) = y ∙ logb(x)For example:

log10(28) = 8∙ log10(2)Derivative of natural logarithmThe derivative of the natural logarithm function is the reciprocal function.When

f (x) = ln(x) The derivative of f(x) is:

f ' (x) = 1 / xIntegral of natural logarithmThe 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) + CLn of 0The natural logarithm of zero is undefined:

ln(0) is undefinedThe limit near 0 of the natural logarithm of x, when x approaches zero, is minus infinity:

Ln of 1The natural logarithm of one is zero:

ln(1) = 0Ln of infinityThe limit of natural logarithm of infinity, when x approaches infinity is equal to infinity:

lim ln(x) = ∞, when x→∞