Presentation on theme: "Ch. 3.2: Logarithms and their Graphs What are they?"— Presentation transcript:
1 Ch. 3.2: Logarithms and their Graphs What are they? Inverse of the exponential functionsDefinition: 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 = 8Ex: log = -4 is the same as 3-4 =Ex: Find x if log4x=2x = ay, sox = 42 = 16
2 Common log function: a = 10 “Log” key on calculator is log, base 10Just enter “log” then the number when working with base 10Calculate1. log log log10-21.73240.3617Not possible
3 PROPERTIES Logarithm (log) loga1=0 because a0 = 1 logaa=1 because a1 = alogaax = x because ax = axlogax = logay, then x = yNatural Log (ln)1. logex = ln x2. y = ln x if and only if x = ey
4 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) = log3x2Ex: f(x) = ln(x – 2) Ex: g(x) = ln x3x2 > 01 – x > 0All numbers workExcept 0!!x < 1x – 2 > 0x3 > 0X > 0x > 2
5 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 calculatorWrite the exponential function y=axFind 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 graphDomain: (0, ∞) Range: (-∞, ∞)X-intercept: (1, 0) V.A.: x = 0 (y-axis)
6 Ex: Graph f(x) = log2x f(x)=log2x f(x)=2x Change to exponential functionCreate a table of valuesFlip the x and y valuesGraph the new set of valuesf(x)=log2xf(x)=2xx-2-112y.25.5124x.25.5124y-2-112y=2xy=log2x*New asymptote is x = 0
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