ELF.01.5 – The Logarithmic Function – Graphic Perspective MCB4U - Santowski

(A) Graph of Logarithmic Functions Graph the exponential function f(x) = 2 x and then graph the inverse y = f -1 (x) which is x = 2 y. This inverse is called a logarithm and is written as y = log 2 (x) Graph the exponential function f(x) = 2 x and then graph the inverse y = f -1 (x) which is x = 2 y. This inverse is called a logarithm and is written as y = log 2 (x) After seeing the graph, we can analyze the features of the graph of the logarithmic function After seeing the graph, we can analyze the features of the graph of the logarithmic function

(A) Graph of Logarithmic Functions

(A) Tables of Values for Exponentials and Logarithmic Functions Exponential Fcns: Exponential Fcns: x y x y   Logarithmic Fcns: x y

(B) Analyzing Graphs of Logarithms So we can now do a complete graphic analysis of this graph So we can now do a complete graphic analysis of this graph (i) no y-intercept and the x-intercept is 1 (i) no y-intercept and the x-intercept is 1 (ii) the y axis is an asymptote (ii) the y axis is an asymptote (iii) range {yER} (iii) range {yER} (iv) domain {x > 0} (iv) domain {x > 0} (v) it increases over its domain (v) it increases over its domain (vi) it is concave down over its domain (vi) it is concave down over its domain (vi) it has no max/min or turning points (vi) it has no max/min or turning points

(C) Graphs of Transformed Logarithmic Functions As will be seen in the next exercises, the graph maintains the same “shape” or characteristics when transformed As will be seen in the next exercises, the graph maintains the same “shape” or characteristics when transformed Depending on the transformations, the various key features (domain, range, intercepts, asymptotes) will change Depending on the transformations, the various key features (domain, range, intercepts, asymptotes) will change

(D) Investigating the Logarithmic Function f(x) = alog B [b(x + c)] + d Go to this link Logarithmic Functions from AnalyzeMath and work through the tutorial on transformed logarithmic functions Go to this link Logarithmic Functions from AnalyzeMath and work through the tutorial on transformed logarithmic functionsLogarithmic Functions from AnalyzeMathLogarithmic Functions from AnalyzeMath Consider how y = log B (x) changes  i.e. the domain, range, asymptotes, increasing/decreasing nature of the function, shifting and reflecting Consider how y = log B (x) changes  i.e. the domain, range, asymptotes, increasing/decreasing nature of the function, shifting and reflecting

(E) Graphing the Logarithmic Function Without Graphing Technology Recall the key features of the logarithmic function, y = log(x): Recall the key features of the logarithmic function, y = log(x): domain {x E R |x > 0} domain {x E R |x > 0} Range is y E R Range is y E R Vertical asymptotes  y axis (or the line x=0) Vertical asymptotes  y axis (or the line x=0) X-intercept  (1,0) X-intercept  (1,0) The function increases over its entire domain The function increases over its entire domain

(E) Graphing Log Functions Without using graphing technology, graph the following functions (it may help to recall your knowledge of function transformations) Without using graphing technology, graph the following functions (it may help to recall your knowledge of function transformations) (1) f(x) = log 2 (x + 2) (1) f(x) = log 2 (x + 2) (2) f(x) = -3log 2 (x - 4) (2) f(x) = -3log 2 (x - 4) (3) f(x) = log 5 (4x – 4) + 5 (3) f(x) = log 5 (4x – 4) + 5 Examples and discussions on how to make these graphs is found at the following website: Examples and discussions on how to make these graphs is found at the following website: Graphs of Logarithmic Functions from AnalyzeMath Graphs of Logarithmic Functions from AnalyzeMath Graphs of Logarithmic Functions from AnalyzeMath Graphs of Logarithmic Functions from AnalyzeMath

(F) Homework Nelson Text, page 118, Q9, 10, 12, 15, 21 Nelson Text, page 118, Q9, 10, 12, 15, 21