ELF.01.5b – The ELF.01.5b – The Logarithmic Function – Logarithmic Function – Algebraic PerspectiveAlgebraic Perspective

MCB4U - SantowskiMCB4U - Santowski

(A) Logarithms as Inverses of (A) Logarithms as Inverses of Exponential FunctionsExponential Functions

if f(x) = aif f(x) = axx, find f , find f -1-1(x) (x) so y = aso y = axx then x = a then x = ayy and now isolate y and now isolate y BUT HOW?? BUT HOW??

in order to isolate the y term, the logarithmic concept was invented in order to isolate the y term, the logarithmic concept was invented or created so we write x = aor created so we write x = ayy as y = log as y = logaa(x) (x)

It is read “y is equal to the logarithm of It is read “y is equal to the logarithm of xx to the base to the base aa””

So notice that:So notice that: The base of the exponent (The base of the exponent (aa) is also the base of the logarithm () is also the base of the logarithm (aa)) The “exponent” of The “exponent” of yy from our exponential equation is now isolated in the from our exponential equation is now isolated in the

logarithmic equationlogarithmic equation In the log form, whatever is “inside” the log (i.e. the In the log form, whatever is “inside” the log (i.e. the xx) is called the ) is called the

argument of the logarithmargument of the logarithm

Notice that a > 0 and a Notice that a > 0 and a ≠ 1 (recall that ≠ 1 (recall that a a is the base of the exponent is the base of the exponent and must be positive and not equal to 1)and must be positive and not equal to 1)

(B) Changing Between Exponential (B) Changing Between Exponential and Logarithmic Formsand Logarithmic Forms

Since x = aSince x = ayy can be rewritten as y = log can be rewritten as y = logaa(x), we can rewrite some (x), we can rewrite some exponential expressions as logarithmic expressions:exponential expressions as logarithmic expressions:

2233 = 8 = 8 can be rewritten can be rewritten 8 = log8 = log2233 10² = 100 10² = 100 can be rewritten can be rewritten 2 = log2 = log1010100100 3344 = 81 = 81 can be rewritten can be rewritten 4 = log4 = log338181

OROR

3 = log3 = log99729 729 can be rewritten can be rewritten 9933 = 729 = 729 -2 = log-2 = log550.04 0.04 can be rewrittencan be rewritten 55-2-2 = 0.04 = 0.04 -3/4 = log-3/4 = log81811/27 1/27 can be rewritten can be rewritten 8181-3/4-3/4 = 1/27 = 1/27

(C) Evaluating Logarithms(C) Evaluating Logarithms We can use our knowledge of exponents to help us evaluate We can use our knowledge of exponents to help us evaluate

logarithmic expressions:logarithmic expressions:

Ex. Evaluate logEx. Evaluate log228 = x8 = x We can convert to exponential form (since this is where logs came We can convert to exponential form (since this is where logs came

from in the first place)from in the first place) So logSo log228 = x becomes 28 = x becomes 2xx = 8 = 8 Thus 2Thus 2xx = 2 = 233 = 8 so x = 3 = 8 so x = 3 Therefore logTherefore log228 = 38 = 3

To consider it another way, remember what logs are used for in the To consider it another way, remember what logs are used for in the first place first place isolating exponents isolating exponents

Therefore, logTherefore, log228 = x is asking us for the exponent of base 2 that 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 gives us the result of 8 (or it is asking us how many times has 2 been multiplied by itself to get an 8) been multiplied by itself to get an 8) which is of course 3 which is of course 3

(C) Evaluating Logarithms(C) Evaluating Logarithms

Ex Evaluate the following:Ex Evaluate the following: loglog4464 = x64 = x loglog991 = x1 = x loglog777 = x7 = x loglog555 = x5 = x loglog64644 = x4 = x loglog44(1/2) = x(1/2) = x loglog44(-16) = x(-16) = x loglog1212(0) = x(0) = x

(D) Common Logarithms(D) Common Logarithms On a calculator, when you use the “log” key, it is given that the base On a calculator, when you use the “log” key, it is given that the base

of the logarithm is 10of the logarithm is 10 log to the base 10 of x (loglog to the base 10 of x (log1010(x)) is called the (x)) is called the common logarithmcommon logarithm of of

xx

If a logarithm is written without a base given, it is implied that the If a logarithm is written without a base given, it is implied that the base is 10base is 10

Ex Ex Evaluate log125 (which would imply log Evaluate log125 (which would imply log1010(125) = x ) (125) = x ) so so what we are looking for is the exponent on 10 that gives us a 125 what we are looking for is the exponent on 10 that gives us a 125 we would use a calculator and get 2.09691 ….. we would use a calculator and get 2.09691 …..

Ex Ex Solve for Solve for xx if log x = 0.25 if log x = 0.25 so the meaning is that 0.25 is the so the meaning is that 0.25 is the exponent on base 10 and I am trying to find out what base 10 raised exponent on base 10 and I am trying to find out what base 10 raised to the exponent 0.25 is to the exponent 0.25 is 10 10(0.25)(0.25) = 1.7783 ….. = 1.7783 …..

(E) Internet Links(E) Internet Links

Tutorial on Logarithmic Functions from WeTutorial on Logarithmic Functions from West Texas A&Mst Texas A&M

A module on Logarithms from A module on Logarithms from PurpleMathPurpleMath

(F) Homework(F) Homework

Nelson text, p117, Q1-5,7,9,12,15 Nelson text, p117, Q1-5,7,9,12,15 Plus some work from HM Math 12Plus some work from HM Math 12