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2. Logarithmic derivative - Wikipedia

   en.wikipedia.org/wiki/Logarithmic_derivative

   Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does not take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have (⁡) ′ = (⁡ + ⁡) ′ = (⁡) ′ + (⁡) ′.

3. List of logarithmic identities - Wikipedia

   en.wikipedia.org/wiki/List_of_logarithmic_identities

   The identities of logarithms can be used to approximate large numbers. Note that log b (a) + log b (c) = log b (ac), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 2 32,582,657 −1. To get the base-10 logarithm, we would multiply 32,582,657 by log 10 (2), getting 9,808,357.09543 ...

4. Logarithmic differentiation - Wikipedia

   en.wikipedia.org/wiki/Logarithmic_differentiation

   It can also be useful when applied to functions raised to the power of variables or functions. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions.

5. Differentiation rules - Wikipedia

   en.wikipedia.org/wiki/Differentiation_rules

   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.

6. Natural logarithm - Wikipedia

   en.wikipedia.org/wiki/Natural_logarithm

   For example, ln 7.5 is 2.0149..., because e 2.0149... = 7.5. The natural logarithm of e itself, ln e, is 1, because e 1 = e, while the natural logarithm of 1 is 0, since e 0 = 1. The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a [4] (with the area being negative when 0 < a < 1 ...

7. Logarithm - Wikipedia

   en.wikipedia.org/wiki/Logarithm

   Moreover, as the derivative of f(x) evaluates to ln(b) b x by the properties of the exponential function, the chain rule implies that the derivative of log b x is given by [35] [37] ⁡ = ⁡. That is, the slope of the tangent touching the graph of the base- b logarithm at the point ( x , log b ( x )) equals 1/( x ln( b )) .