F.4數學3 題

1.
　 3^(2x) = 2^(3y)
log 3^(2x) = log 2^(3y)
　 xlog3² = ylog2³
　　　x/y = log8/log9
　　 x : y = log8 : log9

2.
　 x^(4logx) = 10
logx^(4 logx) = log10
　4logx logx = 1
　　 (logx)² = 1/4
　　　 logx = 1/2　or　logx = -1/2
　　　　　x = 10^(1/2)　or　x = 10^(-1/2)　(rejected as x > 1)

3.
logx + 2 logy = 2 ......(i)
　　　3x - 2y = 2 ......(ii)
From (ii),
　2y = 3x - 2
　 y = (3x - 2)/2 ......(iii)
Sub. (iii) into (i),
　logx + 2 log[(3x - 2)/2] = 2
　 logx + log[(3x - 2)/2]² = 2
　logx + log[(3x - 2)² / 4] = 2
　　　 log[x(3x - 2)² / 4] = 2
　　　　　 x(3x - 2)² / 4 = 10²
　　　　x(9x² - 12x + 4) = 400
　　　　 9x³ - 12x² + 4x = 400
　　9x³ - 12x² + 4x - 400 = 0

Let f(x) = 9x³ - 12x² + 4x - 400
f(4) = 9(4)³ - 12(4)² + 4(4) - 400 = 0
∴ (x - 4) is a factor of f(x).

(x - 4)(9x² + 24x + 100) = 0
　　　　　　　　 x - 4 = 0　or　9x² + 24x + 100 = 0 (rejected)
　　　　　　　　　　x = 4
Sub. x = 4 into (iii),
y = (12 - 2)/2
　= 5
∴ x = 4, y = 5

2007-01-16 16:51:50 補充：
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計住一題ans.:
x^(4logx) = 10
logx^(4 logx) = log10
　4logx logx = 1
　　 (logx)² = 1/4
　　　 logx = 1/2　or　logx = -1/2
　　　　　x = 10^(1/2)　or　x = 10^(-1/2)　(rejected as x > 1)