Ch. 3.2: Logarithms and their Graphs What are they? Inverse of the exponential functions Definition: For x > 0, a > 0, and a ≠ 1, y = logax if and only if ay = x ***y = logax is equivalent to ay = x, thus a logarithm is an exponent *** Ex: log28 = 3 is the same as 23 = 8 Ex: log3 = -4 is the same as 3-4 = Ex: Find x if log4x=2 x = ay, so x = 42 = 16

Common log function: a = 10 “Log” key on calculator is log, base 10 Just enter “log” then the number when working with base 10 Calculate 1. log1054 2. log102.3 3. log10-2 1.7324 0.3617 Not possible

PROPERTIES Logarithm (log) loga1=0 because a0 = 1 logaa=1 because a1 = a logaax = x because ax = ax logax = logay, then x = y Natural Log (ln) 1. logex = ln x 2. y = ln x if and only if x = ey

Finding the domain of logax Take the x-value set it x > 0 and solve **REMEMBER THE DOMAIN AFFECTS WHERE THE GRAPH WILL OCCUR** Ex: f(x) = log4(1-x) Ex: g(x) = log3x2 Ex: f(x) = ln(x – 2) Ex: g(x) = ln x3 x2 > 0 1 – x > 0 All numbers work Except 0!! x < 1 x – 2 > 0 x3 > 0 X > 0 x > 2

Domain: (0, ∞) Range: (-∞, ∞) X-intercept: (1, 0) V.A.: x = 0 (y-axis) Graphing logs **IMPORTANT: Log functions are the inverses of exponential funcs. **reflected about the line y = x** Easiest method for graphing without a calculator Write the exponential function y=ax Find its ordered pairs, then switch the pairs. Multiply y*coefficient (if any) then plot points. Apply any shifts to these pts. then draw curve/asymptote. **Shifts can occur just like exponential functions, but remember that the domain can be limited** Parent graph Domain: (0, ∞) Range: (-∞, ∞) X-intercept: (1, 0) V.A.: x = 0 (y-axis)

Ex: Graph f(x) = log2x f(x)=log2x f(x)=2x Change to exponential function Create a table of values Flip the x and y values Graph the new set of values f(x)=log2x f(x)=2x x -2 -1 1 2 y .25 .5 1 2 4 x .25 .5 1 2 4 y -2 -1 1 2 y=2x y=log2x *New asymptote is x = 0