UNIT 5 WORKSHEET 7
Properties of Logarithms
The following properties serve to expand or condense a logarithm or logarithmic expression so
it can be worked with.
Properties of logarithms
Example
log a mn = log a m + log a n
log 4 3 x = log 4 3 + log 4 x
m
= log a m − log a n
n
log a m n = n log a m
log 2
Properties of Natural Logarithms
Example
ln mn = ln m + ln n
m
ln = ln m − ln n
n
ln m n = n ln m
ln ( x + 1)( x − 5 ) = ln ( x + 1) + ln ( x − 5 )
log a
x +1
= log 2 ( x + 1) − log 2 5
5
3
log 3 ( 2 x + 1) = 3log 3 ( 2 x + 1)
x
= ln x − ln 2
2
ln 73 = 3ln 7
ln
These properties are used backwards and forwards in order to expand or condense a
logarithmic expression. Therefore, these skills are needed in order to solve any equation
involving logarithms. Logarithms will also be dealt with in Calculus. If a student has a firm
grasp on these three simple properties, it will help greatly in Calculus.
Expanding Logarithmic Expressions
Write each of the following as the sum or difference of logarithms. In other words, expand
each logarithmic expression.
A) log 2
3x3 y 2
z5
B) log 3 5 3 xy 2
3
C) log 4 ( x + 1) ( x − 2 )
5
E) log 2
3( x + 2)
x −1
G) log a 12 x
3
y
2
D) log 5
6x2
11 y 5 z
F) log12
x−7
x+2
3
H) log 3
5x5 y3
3
z2
Condensing Logarithmic Expressions
Rewrite each of the following logarithmic expressions using a single logarithm. Condense
each of the following to a single expression. Do not multiply out complex polynomials. Just
3
leave something like ( x + 5 ) alone.
1
B) 2 log x + log y
2
A) 3log 4 x − 5log 4 y + 2log 4 z
1
1
2
log 6 + log x + log y
3
3
3
D)
3
1
log 3 16 − log 3 x 3 − 2 log 3 y
4
3
E) 3log 5 x + 2 log 5 y + log 5 z + 2
F)
1
2
log 2 x + log 2 y − 3
3
3
G) log 3 ( x + 2 ) + log 3 ( x − 2 ) − log 3 ( x + 4 )
H)
2
1
1
log ( x + 1) + log ( x − 2 ) − log ( x + 5 )
3
3
3
C)
Practice Using Properties of Logarithms
Use the following information, to approximate the logarithm to 4 significant digits by using
the properties of logarithms.
log a 2 ≈ 0.3562,
A) log a
6
5
D) log a 30
G) log a
4
9
log a 3 ≈ 0.5646,
and
log a 5 ≈ 0.8271
B) log a 18
C) log a 100
E) log a 3
F) log a 75
H) log a 3 15
I) log a 542