Logarithm - Wikipedia ~ Top Stories Info

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Natural logarithm rules - ln(x) rules

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

Definition of natural logarithm

When

e y = x

Then base e logarithm of x is

ln(x) = loge(x) = y

The e constant or Euler's quantity is:

e ≈ 2.71828183

Ln as inverse function of exponential function

The herbal logarithm function ln(x) is the inverse function of the exponential function ex.

For x>0,

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

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

Natural logarithm rules and homes

Rule title 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 by-product f (x) = ln(x) ⇒ f ' (x) = 1 / x ln integral ∫ ln(x)dx = x ∙ (ln(x) - 1) + C ln of destructive quantity ln(x) is undefined when x ≤ 0 ln of zero ln(0) is undefined ln of 1 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.

logb(x ∙ y) = logb(x) + logb(y)

For instance:

log10(3 ∙ 7) = log10(3) + log10(7)

Logarithm quotient rule

The logarithm of the division of x and y is the adaptation 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 rule

The logarithm of x raised to the ability of y is y occasions the logarithm of x.

logb(x y) = y ∙ logb(x)

For instance:

log10(28) = 8∙ log10(2)

Derivative of herbal logarithm

The derivative of the herbal logarithm function is the reciprocal serve as.

When

f (x) = ln(x)

The by-product of f(x) is:

f ' (x) = 1 / x

Integral of herbal logarithm

The integral of the natural logarithm function is given by means of:

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 close to 0 of the natural logarithm of x, when x approaches zero, is minus infinity:

Ln of one

The herbal logarithm of 1 is zero:

ln(1) = 0

Ln of infinity

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

lim ln(x) = ∞, when x→∞

Complex logarithm

For complicated number z:

z = reiθ = x + iy

The complicated logarithm will probably 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 outlined for actual non certain values of x:

Natural logarithms table

x ln x 0 undefined 0+ - ∞ 0.0001 -9.210340 0.001 -6.907755 0.01 -4.605170 0.1 -2.302585 1 0 2 0.693147 e ≈ 2.7183 1 3 1.098612 4 1.386294 5 1.609438 6 1.791759 7 1.945910 8 2.079442 9 2.197225 10 2.302585 20 2.995732 30 3.401197 40 3.688879 50 3.912023 60 4.094345 70 4.248495 80 4.382027 90 4.499810 100 4.605170 200 5.298317 300 5.703782 400 5.991465 500 6.214608 600 6.396930 700 6.551080 800 6.684612 900 6.802395 1000 6.907755 10000 9.210340

Rules of logarithm ►