Logarithmic Functions

Logarithmic Functions

1. Inverse of Exponential Function

   • a^x = b
     • (base)(exponent) = (output)

2. Loga b = x

   • Log(base) (output) = (exponent)
   • x = loga b, if and only if y = bx. b != 1, b > 0
   • Common Logs: Logs with base = 10
     • don't have apparent base
     • log -1 = undefined
     • When logs have bases different than 10, use the change of base formula
       • Logx y = Log x / Log y
   • Natural logs: ln e x
   • Inverse properties of logs
     • logb b^x = x
     • b^logb x = x
     • ln e^x = x
     • e^ln x = x

3. Theorems

   • logb 1 = 0
   • logb b = 1
   • logb (MN) = logb M + logb N
   • logb (M/N) = logb M - logb N
   • logb m^k = k * logb m
   • log10 10^k = k

4. Graphing Log functions

   • Calculate the values of an exponential function and interchange the values of x and y
   • f(x) = logb (x)
     • Key feature: (1, 0) Asymptote x = 0
     • Horizontal movement f(x) = logb (x +/- h) (opposite sign)
     • Vertical movement f(x) = logb (x) +/- k (same as the sign)
     • Domain (h, ∞) Range (-∞, ∞)
     • Horizontal Reflections & Dilations
       • f(x) = logb(ax)
         • a = negative, reflects over the y-axis
         • |a| > 1 compression
         • 0 < |a| < 1 stretch
     • Vertical Reflections & Dilations
       • f(x) = alogb(x)
         • a = negative, reflection over the x-axis
         • |a| > 1 stretch
         • 0 < |a| < 1 compression