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In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, [1] () ′ = ′ ′ = () ′.
More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor: (/) ′ / = (′ ′) / / = ′ ′, just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.
ln(r) is the standard natural logarithm of the real number r. Arg(z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg(x + iy) = atan2(y, x). Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
In contrast, also shown is a picture of the natural logarithm function ln(1 + x) and some of its Taylor polynomials around a = 0. These approximations converge to the function only in the region −1 < x ≤ 1 ; outside of this region the higher-degree Taylor polynomials are worse approximations for the function.
Computing f'(x) by means of the derivative of ln(f(x)) is known as logarithmic differentiation. [38] ... The logarithm keys (LOG for base 10 and LN for base ...
As x goes to infinity, ψ(x) gets arbitrarily close to both ln(x − 1 / 2 ) and ln x. Going down from x + 1 to x , ψ decreases by 1 / x , ln( x − 1 / 2 ) decreases by ln( x + 1 / 2 ) / ( x − 1 / 2 ) , which is more than 1 / x , and ln x decreases by ln(1 + 1 / x ) , which is less than ...
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459. [1] The natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x.