Logarithmic functions > Logarithmic scales
12345Logarithmic scales

Solutions to the exercises

Exercise 1
a

A ( t ) = 80000 1.06 t

b

Do it.

c

Estimate: about 190000 , the calculator gives you A ( 15 ) 191725 .

Exercise 2
a

See table.

t 0 1 2 3 4 5 6
log(N) 1.70 1.92 2.15 2.37 2.60 2.83 3.05
b

Yes, you get a straight line that goes through ( 0 ; 1.70 ) and ( 4 ; 2.60 ) ) . Since the graph of log ( N ( t ) ) is approximated by a straight line, then N ( t ) must be an exponential function (roughly).

c

log ( N ) 1.70 + 0.22 t

d

N ( t ) 10 1.70 + 0.22 t = 10 1.70 ( 10 0.22 ) t 50 1.66 t

Exercise 3
a

For V ( t ) = b g t the following is true: V ( 0 ) = 2 = b g 0 and V ( 5 ) = 6 = b g 5 .
This results in: b = 2 and g 5 = 6 2 = 3 , and therefore g 1.25 . A suitable formula is V ( t ) 2 1.25 t .

b

V ( t ) = 10 gives you 1.25 t = 5 and therefore t = 1.25 log ( 5 ) 7.21 .

c

V ( t ) = 1 gives you 1.25 t = 0.5 and therefore t = 1.25 log ( 0.5 ) 3.11 .

Exercise 4
a

The tick labels on both axes are powers of 10 .

b

log ( m ) 1.1 and log ( P ) 2.4 .

c

log ( P ) = a log ( m ) + b through ( 1.1 ; 2.4 ) and ( 2.9 ; 2.0 ) .
This results in a = -0.4 1.8 -0.22 and b 2.64 , so log ( P ) -0.22 log ( m ) + 2.64 .

d

P 10 -0.22 log ( m ) + 2.64 = ( 10 log ( m ) ) -0.22 10 2.64 440 m -0.22 .

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