ELF.01.6 - Laws of
Logarithms
MCB4U - Santowski
(A) Review
► if
f(x) = ax, find f -1(x) so y = ax then x = ay
and now isolate y
► in
order to isolate the y term, the
logarithmic notation or symbol or function
was invented or created so we write x = ay
as y = loga(x)
(B) Properties of LogarithmsLogarithms of Powers
►
►
►
►
►
Consider this approach here in
column 1
evaluate log5(625) = x
5x = 625
5x = 54
x=4
►
Now try log5(625) = x tis way:
►
log5(54) = x
log5(5)4 = x
►
►
Now I know from column 1 that
the answer is 4 … so …
►
log5(5)4 = 4
4 x log5(5) = 4
4x1=4
►
►
(B) Properties of LogarithmsLogarithms of Powers
► So
if log5(625) = log5(5)4 = 4 x log5(5)
► It
would suggest a rule of logarithms 
► logb(bx) = x
► Which
we can generalize 
► logb(ax) = xlogb(a)
(C) Properties of Logarithms – Logs
as Exponents
log3 5
► Consider the example 3
x
►
►
►
►
Recall that the expression log3(5) simply means “the
exponent on 3 that gives 5”  let’s call that x
So we are then asking you to place that same exponent
(the x) on the same base of 3
Therefore taking the exponent that gave us 5 on the base
of 3 (x ) onto a 3 again, must give us the same 5!!!!
We can demonstrate this algebraically as well
(C) Properties of Logarithms – Logs
as Exponents
► Let’s
take our exponential equation and write
it in logarithmic form
log3 5
► So
3
x
becomes log3(x) = log3(5)
► Since
both sides of our equation have a log3
then x = 5 as we had tried to reason out in
the previous slide
► So
we can generalize that
b
logb x
x
(D) Properties of Logarithms –
Product Law
►
Express logam + logan as a single logarithm
►
►
We will let logam = x and logan = y
So logam + logan becomes x + y
►
But if logam = x, then ax = m and likewise ay = n
►
►
Now take the product (m)(n) = (ax)(ay) = ax+y
Rewrite mn=ax+y in log form  loga(mn)=x + y
►
►
But x + y = logam + logan
So thus loga(mn) = logam + logan
(E) Properties of Logarithms –
Quotient Law
► We
could develop a similar “proof” to the
quotient law as we did with the product law
 except we ask you to express logam logan as a single logarithm
► As
it turns out (or as is predictable), the
simplification becomes as follows:
► loga(m/n) = loga(m) – loga(n)
(F) Summary of Laws
Logs as
exponents
b
logb x
x
Product
Rule
loga(mn) = logam + logan
Quotient
Rule
loga(m/n) = loga(m) – loga(n)
Power Rule
Loga(mp) = (p) x (logam)
(G) Examples
►
(i) log354 + log3(3/2)
(ii) log2144 - log29
(iii) log30 + log(10/3)
►
(iv) which has a greater value
►
►
 (a) log372 - log38 or (b) log500 + log2
►
(v) express as a single value
 (a) 3log2x + 2log2y - 4log2a
 (b) log3(x+y) + log3(x-y) - (log3x + log3y)
►
►
(vi) log2(4/3) - log2(24)
(vii) (log25 + log225.6) - (log216 + log39)
(G) Internet Links
► Logarithm
► College
Rules Lesson from Purple Math
Algebra Tutorial on Logarithmic
Properties from West Texas AM
(H) Homework
► Nelson
textbook, p125, Q1-14