5.5 Properties of Logarithms
Can you do this in your head? 100 x 1000 = ??
Recall: if m = 102 then 2 = log m ***
100000 =
mn
=
100
m
105
10log mn
10log mn
102 x 103
10log m x 10log n (by ***)
10log m + log n (laws of exponents)
=
=
=
x 1000
x n
log mn = log m + log n (if 10x = 10y then x = y)
the log of a product = the sum of the logs
=========================================
100
= 100000  1000
m/n
=
m
 n
102
10log m/n
10log m/n
=
=
=
105
 103
10log m
 10log n (by ***)
10log m - log n (laws of exponents)
log m/n = log m - log n (if 10x = 10y then x = y)
the log of a quotient = the difference of the logs
==========================================
log mr = log mm … m (for r factors)
= log m + log m + . . . + log m (for r terms)
= r log m
r
log m = r log m
the log of a power = the exponent times the log of the base
==========================================
5.5-1
These relationships hold for any base:
1. loga mn
2. loga m/n
3. loga mr
=
=
=
loga m + loga n
loga m - loga n
r loga m
(log of a product)
(log of a quotient)
(log of a power)
Each property can be used in two directions, e.g.
 log (10)(20) = log 10 + log 20
 uses property 1 going from left-hand to right-hand side
 called expansion ()
 log 10 - log 20 = log (10/20)
 uses property 2 going from right-hand to left-hand side
 called collection, or writing as the log of a single
expression “log expression” ()
 3 log x not acceptable as an answer, but log x3 is
Examples:
1. Expand log2 (4.16)
log2 (4.16) = log2
+log2
(log of a product )
2. Write as the log of a single expression:
log10 0.01 + log10 1000
log10 0.01 + log10 1000 = log10
(log of a product )
3. Expand: loga
loga
4
5 = loga 5
4
5
=
loga 5
(log of a power )
4. Write as the log of a single expression: 5 log 100
5 log 100 = log 100
(log of a power )
5.5-2
x2 y3
5. Expand: loga 4
z
= loga
+ loga
- loga
=
loga x +
loga y loga z
(log of product, quotient, and power )
b
6. Write as a single log: loga
+ loga bx
x
loga
= loga
= loga b(b
) = loga
Change of base formula
 suppose want to find log3 75
 calculator doesn’t have logs to base 3
 not to worry!
 use the change of base formula:
log10 M
logb M =
(that base 10 could be any base)
log10 b
So:
log 75
log3 75 =
= 1.875/.477 = 3.93
log 3
Why does this work? I’ll show you why!
5.5-3