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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] () ′ = ′ ′ = () ′.
The logarithmic derivative is then / and one can draw the general conclusion that for f meromorphic, the singularities of the logarithmic derivative of f are all simple poles, with residue n from a zero of order n, residue −n from a pole of order n. See argument principle. This information is often exploited in contour integration.
For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d. Derivations also use the log definitions x = b log b (x ...
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions. A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al. [ 4 ] See Lützen's scientific bibliography for a sketch of Liouville's original proof [ 5 ...
The logarithmic derivative provides a simpler expression of the last form, as well as a direct proof that does not involve any recursion. The logarithmic derivative of a function f, denoted here Logder(f), is the derivative of the logarithm of the function.
A single branch of the complex logarithm. The hue of the color is used to show the argument of the complex logarithm. The brightness of the color is used to show the modulus of the complex logarithm. The real part of log(z) is the natural logarithm of | z |. Its graph is thus obtained by rotating the graph of ln(x) around the z-axis.
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 ...