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Published byLarissa Poland Modified over 2 years ago

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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 = log a x if and only if a y = x ***y = log a x is equivalent to a y = x, thus a logarithm is an exponent *** Ex: log 2 8 = 3 is the same as 2 3 = 8 Ex: log 3 = -4 is the same as 3 -4 = Ex: Find x if log 4 x=2 x = 4 2 = 16x = a y, so

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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. log log log Not possible

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PROPERTIES Logarithm (log) log a 1=0 because a 0 = 1 log a a=1 because a 1 = a log a a x = x because a x = a x log a x = log a y, then x = y Natural Log (ln) 1. log e x = ln x 2. y = ln x if and only if x = e y

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Finding the domain of log a x Take the x-value set it x > 0 and solve **REMEMBER THE DOMAIN AFFECTS WHERE THE GRAPH WILL OCCUR** Ex: f(x) = log 4 (1-x)Ex: g(x) = log 3 x 2 Ex: f(x) = ln(x – 2)Ex: g(x) = ln x 3 1 – x > 0 x < 1 x 2 > 0 All numbers work Except 0!! x 3 > 0 X > 0 x – 2 > 0 x > 2

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Graphing logs **IMPORTANT: Log functions are the inverses of exponential funcs. **reflected about the line y = x** Easiest method for graphing without a calculator 1.Write the exponential function y=a x 2. Find its ordered pairs, then switch the pairs. 3.Multiply y*coefficient (if any) then plot points. 4.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)

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Ex: Graph f(x) = log 2 x 1.Change to exponential function 2.Create a table of values 3.Flip the x and y values 4.Graph the new set of values f(x)=2 x x y f(x)=log 2 x y x y=2 x y=log 2 x *New asymptote is x = 0

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