ELF.01.5b – The Logarithmic Function – Algebraic Perspective MCB4U - Santowski

(A) Logarithms as Inverses of Exponential Functions  if f(x) = a x, find f -1 (x)  so y = a x then x = a y and now isolate y  BUT HOW??  in order to isolate the y term, the logarithmic concept was invented or created so we write x = a y as y = log a (x)  It is read “y is equal to the logarithm of x to the base a”  So notice that: The base of the exponent (a) is also the base of the logarithm (a) The base of the exponent (a) is also the base of the logarithm (a) The “exponent” of y from our exponential equation is now isolated in the logarithmic equation The “exponent” of y from our exponential equation is now isolated in the logarithmic equation In the log form, whatever is “inside” the log (i.e. the x) is called the argument of the logarithm In the log form, whatever is “inside” the log (i.e. the x) is called the argument of the logarithm  Notice that a > 0 and a ≠ 1 (recall that a is the base of the exponent and must be positive and not equal to 1)

(B) Changing Between Exponential and Logarithmic Forms  Since x = a y can be rewritten as y = log a (x), we can rewrite some exponential expressions as logarithmic expressions:  2 3 = 8 can be rewritten  8 = log 2 3  10² = 100 can be rewritten  2 = log  3 4 = 81 can be rewritten  4 = log 3 81  OR  3 = log can be rewritten  9 3 = 729  -2 = log can be rewritten  5 -2 = 0.04  -3/4 = log 81 1/27 can be rewritten  81 -3/4 = 1/27

(C) Evaluating Logarithms  We can use our knowledge of exponents to help us evaluate logarithmic expressions:  Ex. Evaluate log 2 8 = x  We can convert to exponential form (since this is where logs came from in the first place)  So log 2 8 = x becomes 2 x = 8  Thus 2 x = 2 3 = 8 so x = 3  Therefore log 2 8 = 3  To consider it another way, remember what logs are used for in the first place  isolating exponents  Therefore, log 2 8 = x is asking us for the exponent of base 2 that gives us the result of 8 (or it is asking us how many times has 2 been multiplied by itself to get an 8)  which is of course 3

(C) Evaluating Logarithms  Ex Evaluate the following: log 4 64 = x log 4 64 = x log 9 1 = x log 9 1 = x log 7 7 = x log 7 7 = x log 5  5 = x log 5  5 = x log 64 4 = x log 64 4 = x log 4 (1/2) = x log 4 (1/2) = x log 4 (-16) = x log 4 (-16) = x log 12 (0) = x log 12 (0) = x

(D) Common Logarithms  On a calculator, when you use the “log” key, it is given that the base of the logarithm is 10  log to the base 10 of x (log 10 (x)) is called the common logarithm of x  If a logarithm is written without a base given, it is implied that the base is 10  Ex  Evaluate log125 (which would imply log 10 (125) = x )  so what we are looking for is the exponent on 10 that gives us a 125  we would use a calculator and get …..  Ex  Solve for x if log x = 0.25  so the meaning is that 0.25 is the exponent on base 10 and I am trying to find out what base 10 raised to the exponent 0.25 is  10 (0.25) = …..

(E) Internet Links  Tutorial on Logarithmic Functions from West Texas A&M Tutorial on Logarithmic Functions from West Texas A&M Tutorial on Logarithmic Functions from West Texas A&M  A module on Logarithms from PurpleMath A module on Logarithms from PurpleMath A module on Logarithms from PurpleMath

(F) Homework  Nelson text, p117, Q1-5,7,9,12,15  Plus some work from HM Math 12