Logarithms 23
= 6 log2 − 7 log2 + 5 log2
by the third law of logarithms
= 4 log2
Problem 15. Write
1
2
log16 +
1
3
log27 − 2 log5
as the logarithm of a single number.
1
2
log 16 +
1
3
log 27 − 2 log5
= log 16
1
2 + log 27
1
3 − log 5
2
by the third law of logarithms
= log
√
16 + log
3
√
27 − log 25
by the laws of indices
= log4 + log 3 − log 25
= log
4 × 3
25
by the first and second laws of logarithms
= log
12
25
= log 0.48
Problem 16. Write (a) log30 (b) log 450 in terms
of log 2, log3 and log5 to any base.
(a) log 30 = log(2 × 15) = log(2 × 3 × 5)
= log 2 + log 3 + log 5
by the first law of logarithms
(b) log 450 = log(2 × 225) = log(2 × 3 × 75)
= log(2 × 3 × 3 × 25)
= log(2 × 3 2 × 5 2 )
= log2 + log 3 2 + log 5 2
by the first law of logarithms
i.e. log 450 = log 2 + 2 log3 + 2 log5
by the third law of logarithms
Problem 17. Write log
8 ×
4
√
5
81
in terms of
log 2, log3 and log5 to any base.
log
8 ×
4
√
5
81
= log 8 + log
4
√
5 − log 81
by the first and second
laws of logarithms
= log 2
3
+ log 5
1
4 − log 3
4
by the laws of indices
i.e. log
8 ×
4
√
5
81
= 3 log2 +
1
4
log 5 − 4 log3
by the third law of logarithms
Problem 18. Evaluate:
log 25 − log125 +
1
2 log 625
3 log5
.
log 25 − log125 +
1
2 log 625
3 log5
=
log 5 2 − log 5 3 +
1
2 log 5 4
3 log5
=
2 log5 − 3 log5 +
4
2 log 5
3 log5
=
1 log5
3 log5
=
1
3
Problem 19. Solve the equation:
log(x − 1) + log(x + 8) = 2 log(x + 2).
LHS = log (x − 1) + log(x + 8)
= log (x − 1)(x + 8)
from the first law of logarithms
= log (x
2
+ 7x − 8)
RHS = 2 log(x + 2) = log (x + 2)
2
from the third law of logarithms
= log(x
2
+ 4x + 4)
log(x
2
+ 7x − 8) = log (x
2
+ 4x + 4)
Hence,
x
2
+ 7x − 8 = x
2
+ 4x + 4
from which,
7x − 8 = 4x + 4
i.e.
3x = 12
i.e.
x = 4
and
Problem 20. Solve the equation:
1
2
log 4 = log x.
1
2
log 4 = log4
1
2
from the third law of logarithms
= log
√
4 from the laws of indices