LOGARITHMS AND EXPONENTIAL FUNCTIONS 25

A

(b) Letx=log 10 10 then 10x=10 from the defin-

ition of a logarithm, i.e. 10x= 101 , from which

x= 1

Hencelog 1010 = 1 (which may be checked by

a calculator)

(c) Letx=log 16 8 then 16x=8, from the defini-

tion of a logarithm, i.e. (2^4 )x= 23 , i.e. 2^4 x= 23

from the laws of indices, from which, 4x=3 and

x=^34

Hencelog 168 =^34

Problem 2. Evaluate (a) lg 0.001 (b) ln e

(c) log 3

1

81

.

(a) Letx=lg 0. 001 =log 100 .001 then 10x= 0 .001,

i.e. 10x= 10 −^3 , from whichx=− 3

Hencelg 0.001=− 3 (which may be checked

by a calculator)

(b) Let x=ln e=logee then ex=e, i.e. ex=e^1
from whichx=1. Henceln e= 1 (which may
be checked by a calculator)

(c) Letx=log 3

1

81

then 3x=

1

81

=

1

34

= 3 −^4 , from

whichx=− 4

Hencelog 3

1

81

=− 4

Problem 3. Solve the following equations:

(a) lgx=3 (b) log 2 x=3 (c) log 5 x=−2.

(a) If lgx=3 then log 10 x=3 andx= 103 , i.e.

x= 1000

(b) If log 2 x=3 thenx= 23 = 8

(c) If log 5 x=−2 thenx= 5 −^2 =

1

52

=

1

25

Problem 4. Write (a) log 30 (b) log 450 in

terms of log 2, log 3 and log 5 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× 32 × 52 )

=log 2+log 3^2 +log 5^2 ,

by the first law of logarithms

i.e. log 450=log 2+2 log 3+2 log 5,

by the third law of logarithms

Problem 5. Write log

(

8 ×^4

√

5

81

)

in terms of

log 2, log 3 and log 5 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 log 2+^14 log 5−4 log 3,
by the third law of logarithms
Problem 6. Evaluate
log 25−log 125+^12 log 625
3 log 5
.
log 25−log 125+^12 log 625
3 log 5

log 5^2 −log 5^3 +^12 log 5^4
3 log 5

2 log 5−3 log 5+^42 log 5
3 log 5

1 log 5
3 log 5

1
3
Problem 7. Solve the equation:
log (x−1)+log (x+1)=2 log (x+2).