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Lecture

logirithms.docx

4 Pages
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Department
Mathematics
Course Code
MATA30H3
Professor
Ken Butler

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Description
Exponential Functions and its Inverse  an exponential function of the form y = bx , b > 0, b not equal 1, has  a repeating pattern of finite differences  a rate of change that is increasing proportional to the function for b > 1  a rate of change that is decreasing proportional to the function for 0 < b < 1  An exponential function of the form y = bx , b > 0, b not equal 1,  has a domain X E R  a range Y E R, Y > 0  a y-intercept of 1  has a horizontal asymptote at y = 0  is increasing on its domain when b > 1  is decreasing on its domain when 0 < b < 1  The inverse of y = bx is a function that can be written as x = by.  has a domain of X E R, x > 0  a range of Y E R  a x-intercept of 1  has vertical asymptote at x = 0  is a reflection of y = bx about the line y = x  is increasing on its domain when b > 1  is decreasing on its domain when 0 < b < 1 Logarithms  a logarithmic function is the inverse of the exponential function  The value of logbx is equal to the exponent to which the base, b, is raised to produce product x  Exponnetial equations can be written in logarithmic form, and vice versa  y = b^x x = logby  y = logbX x = b^y  Exponential and logarithmic functions are defined only for positive values of the base that are not equal to one. In other words, b not = 1, and x > 0.  The logarithm of x to base 1 is only valid when x = 1, in which base y has an infinite number of solutions and is not a function.  Common logarithms are logarithms wit ha base of 10. It is not necessary to write the base for common logarithms: logx means log base 10 x. Transformations of logarithmic Functions  The techniques for applying transformations to logarithmic functions are the same for those used for other functions:  y = log x + c  translate up c units if c > 0  translate down c units if c < 0  y = log( x – d)  translate right d units if d > 0  translate left d units if d < 0  y = a log x  stretch vertically by a factor of |a| if |a| > 1  Compress vertically by a factor of |a| if |a| < 1  Reflect in the x-axis if a < 0  y = log (kx)  compress horizontally by a factor of |1/k| if |k| < 1, k not = 0.  Reflect in the y axis if k < 0.  When all transformations are combined, they follow the form: f(x) = a log[k(x-d)] + c Power of logarithms
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