Intro to Logarithms Worksheet
Properties of Simple Logarithms
log
log 1 0
log 1
log ( )
log log
a
a
a
x
x
a
aa
a
a x and a x inverse property
If x y then x y
=
=
==
==
Properties of Natural Logarithms
ln
ln1 0
ln 1
ln ( )
ln ln
xx
e
e x and e x inverse property
If x y then x y
=
=
==
==
A standard logarithm can have any positive number as its base except 1, whereas a natural log is always base
e
. Since the natural log is always base
e
, it will be necessary to use a calculator to evaluate natural logs
unless one of the first three examples of the properties of natural logs is used. For anything such as
ln2=
, a
calculator must be used.
When dealing with logarithms, switching between exponential and Logarithmic form is often necessary.
Logarithmic form
Exponential Form
log
a
bc=
c
ab=
Write each of the following in exponential form.
1)
4
log 16 2=
2)
9
1
log 3
2
=
3)
9
3
log 27
2
=
4)
4
1
log 2
16
=−
Write each of the following in logarithmic form.
5)
4
3 81=
6)
14
16 2=
7)
12
1
36
6
−
=
8)
54
16 32=
Simplifying Logarithms
Evaluate each of the following logarithms without the use of a calculator. Remember to write in
exponential form to help if needed.
9)
3
log 81=
10)
4
1
log
2
=
11)
12
log 144=
12)
6
1
log
36
=
13)
2
3
9
log
4
=
14)
0.25
log 4 =
15)
3
log 3−=
16)
8
log 4 =
17)
81
1
log
27
=
18)
1
16
log 32 =
19)
4
log 0 =
20)
10
log 1=
21)
4
1
log
8
=
22)
27
1
log
3
=
23)
9
log 3=
24)
3
6
log 6
x
=
25)
36
1
log
6
=
26)
128
log 2=
27)
1
4
log 16 =
28)
2
log
x
z
z =
29)
12
lne =
30)
3
log 5
3 =
31)
ln1=
32)
ln4x
e =
