急! Logerithmic Functions 9q10

25.
Io : intensity of light
I : intensity of light when the diver goes down n metres

I = Io*(1 - 2.5%)ⁿ

(a)
When I = (1/4)*Io :
(1/4)*Io = Io*(1 - 2.5%)^n.... [^n : to the power n]
0.25 = 0.975^n
log 0.25 = n log 0.975
n = log 0.25 / log 0.975
n ≈ 55

Depth that the diver should go down = 55 m

(b)
When I = 1%*Io :
1%*Io = Io*(1 - 2.5%)^n
0.01 = 0.975^n
log 0.01 = n log 0.975
n = log 0.01 / log 0.975
n ≈ 182 (to the nearestinteger)

Depth that the diver should go down = 182 m


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26.
N = 74 - 50 log x

Yesterday, N = No and x = xo :
No = 74 - 50 log xo ...... [1]

Today, x = (xo + 5) and N = No - 7 :
No - 7 = 74 - 50 log (xo + 5) ...... [2]

[1] - [2] :
7 = 50 log (xo + 5) - 50 log xo
log[(xo + 5)/xo] = 0.14
[(xo + 5)/xo] = 10^0.14
xo + 5 = xo * 10^0.14
xo (10^0.14 - 1) = 5
xo ≈ 13 (to the nearest dollar)

The price of the first shrimps per 100 g yesterday = $13


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27.
Richter magnitudes : M = log A - log Ao
where :
M : magnitude of the earthquake
A : amplitude of the earthquake
Ao : reference amplitude

When A = a unit, M = 7 :
7 = log a - log Ao ...... [1]

When A = a/3000, M = Mx :
Mx = log (a/3000) - log Ao ...... [2]

[1] - [2] :
7 - Mx = log a - log(a/3000)
7 - Mx = log a - (log a - log 3000)
Mx = 7 - log 3000
Mx ≈ 3.5 (to 1 decimal place)

The magnitude of the earthquake in city B = 3.5