Chapter 4: Logarithms and Exponents

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Law of exponents

a^m * a^n = a^(m+n)

a^m / a^n = a^(m-n)

(a^m)^n = a^(mn)

a^0 = 1

n√ a = a^(1/n)

n√ (a^m) = (n√ a)^m = (a^m)^(1/n) = (a^1/n)^(m) = a^(m/n)

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Exponential functions

  • An exponential function is a function of the form f(x) = a^x when a is a positive real number (that is, a>0) and a ≠ 1.
  • The domain of the exponential function is the set of all positive real numbers
  • The range of the exponential function is the set of all positive real numbers
  • The graph of the exponential function f(x) = e^(-x) is a graph of exponential decay
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Logarithms properties

a=b^c then c=logb(a)

loga(a) = 1

loga(a) = 0

loga(b) is undefined for any negative base.

loga(0) is undefined

loga(a^n) = n

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Logarithmic functions

  • Generally is f:x → a^x then f^-1:x→loga(x)
  • y=loga(x) is the inverse of y=a^x
  • y=Ln(x) is the inverse of the exponential function y=e^x
  • loga(a^x) = x and a^(loga(x)) = x
  • Ln(e^x) = x and e^(Ln(x)) = x
  • log(10x) = x and (10^log(x)) = x

 

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Laws of logarithms

  • log(x) + log(y) = log(xy)
  • log(x) - log(y) = log(x/y)
  • log(x'^n) = nlog(x)
  • log(1/x) = -log(x)

 

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Change of base formula

  • logb(a) = (logc(a))/(logc(b))
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