Logarithmic functions > Properties
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Solutions to the exercises

Exercise 1
a

10 log ( 5 20 ) = 10 log ( 100 ) = 2

b

5 log ( 100 / 4 ) = 5 log ( 25 ) = 2

c

6 log ( 3 2 4 ) = 6 log ( 36 ) = 2

d

1 3 log ( 45 / 5 ) = 1 3 log ( 9 ) = -2

Exercise 2
a

5 log ( 5 4 ) = 4

b

2 log ( 100 ) = log ( 100 ) log ( 2 ) 6.644

c

7 log ( 7 0.5 ) = 0.5

d

8 log ( 8000 ) = log ( 8000 ) log ( 8 ) 4.322

e

log ( 50 ) log ( 1 / 3 ) -3.561

f

log ( 40 25 ) = log ( 10 3 ) = 3

g

log ( 0.0003 ) log ( 1 / 3 ) 7.384

Exercise 3
a

0.93 t = 0.5 , therefore t = 0.93 log ( 0.5 ) 9.55 years.

b

400 200 100 50 , so 3 half-times, and that is 3 9.55 = 28.65 years.

c

50 0.93 t = 10 , so 0.93 t = 0.2 and t = 0.93 log ( 0.2 ) 22.18 years.

Exercise 4
a

2 half-times, so 2 165 = 330 days.

b

3 half-times, so 495 days.

c

100 50 25 12,5 , so a little less than 495 days.

d

g ( 165 ) = 0.5 , so g day 0.9958 .
100 0.9958 t = 15 , therefore 0.9958 t = 0.15 and t = 0.9958 log ( 0.15 ) 451 days.

Exercise 5

The growth percentage is p , so the growth factor is 1 + p 100 = g .
The doubling time T is defined as g T = 2 , so ( 1 + p 100 ) T = 2 . This gives you log ( ( 1 + p 100 ) T ) = log ( 2 ) , so T log ( 1 + p 100 ) = log ( 2 ) and from that you can derive the above formula.

Exercise 6
a

5 x = 0.016 gives you x = 5 log ( 0.016 ) -2.6 .

b

x 2 = 3 3 = 27 gives you x = ± ( 27 ) .

c

log ( 2 x ) - log ( x 2 ) = log ( 2 x ) = 1 gives you 2 x = 10 1 = 10 and therefore x = 2 10 = 0.2 .

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