WebbWe have with u = e−x so du = −e−xdx dx = − udu ∫ 1+e−xdx = −∫ u(1+u)du = ∫ 1+udu − ∫ udu = ln(1+u)− lnu+ C = ln(1+ ex)+C. For 1,4,6 you can use the derivative. f 1′(x)= 3x2 +1 > 0 f 4′(x) = 1+ 1+x21 > 0 f 6′(x) = 3+cos(x) > 0 these functions are continuous and strictly increasing, thus they have an inverse ... WebbProve the concavity of F (x,y) = ln(x)+ y by arguing the definition of concavity. You need to show that for any x0,y0 and x1,y1 and any t ∈ [0,1], ln(tx0 +(1− t)x1) +ty0 + (1− t)y1 ≥ …
Explanation for equivalence of ln 1/2 - Mathematics Stack Exchange
Webb3 dec. 2015 · Explanation: Since ln(x) and ex are inverse functions, ln(ex) = x for all values of x. Since 1 e3 = e−3 by definition of negative exponents, it follows that ln( 1 e3) = ln(e−3) = −3. You could also note that ln( 1 e3) = ln(1) − ln(e3) = 0 − 3 = − 3 since ln(A B) = ln(A) − ln(B) and ln(1) = 0. Answer link. Webb1 mars 2024 · -1 Division rule of logarithms states that: ln(x/y) = ln(x) - ln(y) Here we can substitute: ln(1/e)=ln(1) - ln(e) 1) Anything to the power 0=1 2) ln(e)=1, as the base of … data collection points in research
Natural logarithm rules - ln(x) rules - RapidTables.com
WebbWhat is the natural logarithm of 1? What is the natural logarithm of one. ln(1) = ? The natural logarithm of a number x is defined as the base e logarithm of x: ln(x) = log e (x) … WebbSolve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. WebbStep 1: Enter the expression you want to simplify into the editor. The simplification calculator allows you to take a simple or complex expression and simplify and reduce … data collection research paper