1

99 16 25
log x log x log x
88 15 24
Using the rules of logarithms
9 15 24
a b c log x
8 16 25
81 1
log x log x1
80 80
Example 5 Given that
4log
64 N p and 2log 416N
qand
q p 8. Find N .
Solution
Let log
log
4 64
N
be
4 4
64
N t N 64
4 3t
log 4 N log 4 4
4
log 4 N 3t
1 4
t log 4 N log
3
t
4
N
4
3
2
16
N log N 8
log 3
log
16N
N
2
N
4
N 3
N
N
4 4
2
16 N
4
N 3
2
4
4
3
2
4
2
3
2
3
N
4
4
4
4
4
4
8
4
8
4
3
4 6
4 2 4 4096
Example 6 Given that
log y w a and log x w b
t
where w 1
log
Prove that
n x log n y a b
log n x log n y a b
Verify your answer without using a calculator
or tables when
y 4,
n 2, x 8 and w 4096
Solution:
log y w a
(a)
w y
log
n
log
a
log
w alog
n
n
w
y
a b log
a b log
log
n
w
n
n
1
log
n
n
w
y
a
y
y
log
log
1
log
n
n
n
w
x
log
n
log
log
log
log
log
x
n
log
x
w b
w x
w blog
n
n
n
n
w
b
x
w log
y log
1
w
log n y
n
n
n
b
x
w
x
1
log
log
=
n x log n y
log x y
n log n
log n y log n x log n y log n x
log n x log n y
log x log y
n
n
Given y = 4, n = 2, x = 8, and w = 4096
a b
log 4 4096 6 ,log 8 4096 4
6 4 log 2 8 log 4
2
6 4 log 8 log 4
2 3 2
10 3 2
1 1
5 5
Hence it is verified
Example 7
2
Solution
Let log y a
a
y x
log
y
x
y a log
1 a log
a
1
log
y
y
y
x
2
Prove that
x
x
log y
x
1
log
y
x
n
x