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The natural logarithm is the limit of the integrals of powers
Beck, George
  • Natural Logarithm, Limit, Integral
  • Assume that a=/0 and that x>0 The integral of x^(a-1) is (x^a)/a + C, where C is an arbitrary constant. The integral of x^-1=1/x is log(x)+C´, where again C' is an arbitrary constant and log(x) is the natural logarithm of x, often written as ln(x) When a is close to zero, x^(a-1) and x^-1are close, so there must be some connection between their integrals! Choose C=-1/a and C´=0 so that the two integrals are both zero at x=1. The integrals are then ((x^a)/a)-(1/a) and log(x). For a close to zero these functions are very close; in symbols, lim ((x^a)/a-(1/a)=log(x) when a tends to 0 Using the difference quotient for the derivative of the base-x exponential function f(b)=x^b with respect to b (not x) and using a instead of the more usual h gives f'(b) = lim(f(a+b)-f(b))/a = lim(x^(b+a)- x^b)/a = x^b lim(x^b-1)/b) = x^b log(x). This is more usually written with x as the variable: (d/dx)u^x=u^zlog(u), with the special case (d/dx)e^x=e^x
  • Componente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemática
Issue Date: 
  • 20-Sep-2008
  • 20-Sep-2008
  • 2008
  • 20-Sep-2008
  • 12-Sep-2008
Wolfram Demonstration Project
  • Natural Logarithm
  • Limit
  • Integral
  • Educação Superior::Ciências Exatas e da Terra::Matemática::Análise Funcional
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Appears in Collections:MEC - Objetos Educacionais (BIOE) - OE

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