Properties Of Logarithms Investigation Sheet
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44.8A – Properties of Logarithms
Through various rearrangements in exponential form we can come up with the following basic
properties to facilitate our evaluation of logarithms.
What exponent do you have to put base ‘5” to get 1?
a) log
1 = 0
ex. log
1 = 0
b
5
What exponent do you have to put base ‘5” to get 5?
b) log
b = 1
ex. log
5 = 1
b
5
x
x
c) log
b
= x
ex. log
5
= x
b
5
This follows from previous example.
x
That is exponent needed to get 5
?
=
=
log
x
log
x
d) b
x
ex. b
x
b
b
In addition to the basic properties the following operational properties prove useful when trying
to rearrange logarithmic expressions.
Exponent moves in front to become multiplier
See
r
3
a) log
x
= r log
x
ex. log
5
= 3 log
5
investigation
a
a
2
2
sheets to see
b) log
xw = log
x + log
w
ex. log
(3)(4) = log
3 + log
4
a
a
a
5
5
5
how these
properites
⎛
⎞
⎛
⎞
x
2
=
−
=
−
log ⎜
⎟
⎜
⎟
c)
log
x
log
w
ex.
log
log
2
log
3
might have
a
a
a
5
5
5
⎝
⎠
⎝
⎠
w
3
developed
Example 1:
Express as single logarithm
a) log
8 + log
30
b) log 150 - log 25
c) 3log (x+3) - 2log (x-1)
5
5
=
+
−
−
= log
240
= log 6
3
2
240=8x3
6=150/25
log(
x
) 3
log(
x
) 1
5
+
3
(
x
) 3
=
log
−
2
(
x
) 1
Example 2:
Simplify the following
⎛
⎞
1
27
−
a)
b)
c)
d)
−
log
49
⎜
⎟
3
log
16
log
log
48
log
3
2
3
3
4
2
2
⎝
⎠
81
=
⎛
⎞
2
48
log
4
−
1
=
−
=
⎜
⎟
4
log
27
log
81
=
2
log
log
49
3
3
3
3
2
=
⎝
⎠
3
2
=
−
3
4
⎛
⎞
log
3
log
3
1
=
⎜
⎟
3
3
log
16
log
3
⎝
⎠
2
=
7
=
−
3
3
4
=
4
log
2
=
−
2
1
1
=
We now have a
=
4
way or rewriting
7
this to solve. One
can memorize the
Example 3:
Solve the following
rearrangement to
come up with base
X
x
a) 7
= 400
b) 7(1.06)
= 5.20
c) log
50
3
change technique
x
x
x
log 7
= log 400
log 7(1.06)
= log 5.2
set
3
= 50
x log 7 = log 400
log 7 + x log 1.06 = log 5.2
x log 3 = log 50
x = log 400 ÷ log 7
x log 1.06 = log 5.2 – log 7
x = log 50 ÷ log 3
x = 3.08
x = (log 5.2 – log 7) ÷ log 1.06
x = 3.56
x = -5.10
Take log base 10 of
both sides. This is
similar to squaring
base 10 now allow
both sides, it is just a
log
50
you to use calculator
=
different operation
x
Written in one line this looks like.
log
3
(or function).
−
log
5
2 .
log
7
=
x
log
. 1
06
4.8A – Properties of Logarithms
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