Chapter 4: Logarithms and 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)

• 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

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

• 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

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

• logb(a) = (logc(a))/(logc(b))