Form4 maths question about logarithmic functions(3)

用戶22906

2008-10-28 3:34 pm

40.Consider 2^x=5^y=10^z, where x,y and z are non-zero real numbers.
(a) Express x,y in terms of z respectively.
(b) Hence show that 1/x+1/y=1/z
41.If x^0.25 +x^(-0.25)=4, find
(a)(√x )+(1/√x)
(b)x+(1/x)
44.The population of a city is four million at the end of the year 204. The population grows at the rate 3% per year.
(a) In which year will the population be doubled?
Ans:
40(a)x= z/log2
y=z/log5
41.(a)14
(b)194
44.(a)2028
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回答 (1)

用戶10012

2008-10-28 6:00 pm

✔ 最佳答案

Q1.
2^x = 10^z
x log 2 = z log 10 = z........(1)
5^y = 10^z
y log 5 = z log 10 = z .........(2)
So 1/x + 1/y = log 2/z + log 5/z = (log 2 + log 5)/z = (log 10)/z = 1/z.
Q2.
[x^0.25 + x^(-0.25)]^2 = x^0.5 + 2(x^0.25)[x^(-0.25)] + x^(-0.5)
= x^0.5 + 2 + x^(-0.25) = 4^2 = 16, therefore,
x^0.5 + x^(-0.5) = 16 - 2 = 14.
Similarly, [x^0.5 + x^(-0.5)]^2 = x + 2 + 1/x = 14^2 = 196, therefore,
x + 1/x = 196 - 2 = 194.
Q3.
Population at 2004 = 4 million.
population n years later = 8 million.
So 8 = 4( 1 + 0.03)^n
2 = (1.03)^n
log 2 = n log 1.03
n = log 2/log 1.03 = 0.301/0.0128= 23.4
So population double in 2004 + 24 = 2028.

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