Section 1.8 Logarithmic functions

Logarithmic functions Definition: Let b > 0 and b not equal to 1. Then y is the logarithm of x to the base be written: y = logbx if and only if by = x

Example 24 = x 3x = 9 logb16 = 4 log5 x = 2

Example 434 = 81 ½361/2 = 6 050 = 1

Basic logarithmic properties Why? b1 = b Why? b0 = 1 Why? bx = bx Also, 𝑏 𝑙𝑜𝑔 𝑏 𝑥 =𝑥

Other logarithms logb1 = 0 log 1 = 0 ln 1 = 0 logbb = 1 log 10 = 1 Common Logarithm: (base 10) log x = y  10y = x Natural Logarithm: (base e) logex = y ln e = y  ey = x General Properties Common Logarithms Natural Logarithms logb1 = 0 log 1 = 0 ln 1 = 0 logbb = 1 log 10 = 1 ln e = 1 Logbbx = x log 10x = x ln e x =x 𝑏 𝑙𝑜𝑔 𝑏 𝑥 =𝑥 10logx = x elnx = x

log216= 2 log8 + logx + logy logb (xy) = logb x + logb y Log of a Product: logb (xy) = logb x + logb y Examples: log216= 2 log4 8 + log4 2 = log8 + logx + logy log 8xy =

Log5125=3 log 5 + log m – log n logb = logb x - logb y Log of a Quotient: logb = ÷ ø ö ç è æ y x logb x - logb y Examples: Log5125=3 log5 375 - log5 3 = log = n m 5 log 5 + log m – log n

7(logr + logt) logb xn = n · logb x Log of a Power: Examples: log (rt)7 =

Use the properties to rewrite equations without logs Consider: log R = 4log x - log 3 logR = logx4 – log3 logR = logx4/3 R = x4/3

log32/log8 = 5/3 log18/log5 = 1.80 Change of base theorem logba = log (a) /log (b) log832 = log518 = Can also use to graph on calculator: y = log2xy = log(x)/log(2) log32/log8 = 5/3 log18/log5 = 1.80

Graphing logarithms: y = log2x To graph y = logb x Algebraically: Rewrite as an exponential equation: by = x Make an x/y table, filling in y first. Graph points. Using the calculator Re-write using change of base theorem Input y1 = log(x)/log(b) x Y ½ -1 1 2 4

Asymptotes (line that graph approaches, but does not touch) Properties of y = logbx y = log bx OR x = by y = bx Domain Range Asymptotes (line that graph approaches, but does not touch) Point on all graphs inverses x > 0 All reals All reals y > 0 x = 0 y = 0 (1, 0) (0, 1)

Example

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