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標題:
Logarithmic equation(hard~help!)
發問:
Solve the following equations, corr. to 2 d.p.~ 1. 3[5^(x+3)] = 4[7^(2x-1)] 2. [6^(x+1)]^2 = 2[3^(2x-1)] 3. (5/2)^(x-1)．(1/2)^x = 25/2 4. 2[6^(2-x)] - 3 =13 Given 8^x = 0.08^y = 1000. Without finding the values of x, y, find the value of 1/x - 1/y. 更新: ANS 1)2.84 2)-2.88 3)15.4 4)0.84 *last Q: 2/3
最佳解答:
1)3(5x+3)=4(72x-1) log3(5x+3)=log4(72x-1) log3+log5x+3=log4+log(72x-1) log4-log3=(x+3)log5-(2x-1)log7 log4/3=xlog5+3log5-2xlog7+log7 log4/3-3log5-log7=(log5-2log7)x x=2.84 (cor. to 2 d.p.) 2)(6x+1)2 = 2(32x-1) 62x+2=2(32x-1) log62x+2=log2(32x-1) (2x+2)log6=log2+log32x-1 2xlog6+2log6=log2+(2x-1)log3 2xlog6=log2-2log6+2xlog3-log3 x(2log6-2log3)=log2-2log6-log3 x=-2.88 (cor. to 2d.p.) 3)(5/2)x-1(1/2)x = 25/2 (5/2)x(2/5)(1/2)x = 25/2 [(5/2)(1/2)]x = 31.25 (5/4)x = 31.25 log (5/4)x = log 31.25 x log (5/4) = log 31.25 x = 15.43 ( cor. to 2 d.p. ) 4)2(62-x) - 3 =13 2(62-x)=16 62-x=8 (36)(6-x)=8 6-x=2/9 1/6x=2/9 6x=4.5 log6x=log4.5 xlog6=log4.5 x=0.84 ( cor. to 2d.p. ) 5)8x = 0.08y = 1000 So 8x = 1000 xlog8 = log1000 x = 3 / log 8 0.08y = 1000 ylog0.08= log1000 y=3/log0.08 = 3/[log(8/100)] = 3/(log8-2) Then, 1/x-1/y = 1/(3/log8)-1/[3/(log8-2)] =log8/3-(log8-2)/3 =(log8-log8+2)/3 =2/3
Logarithmic equation(hard~help!)
發問:
Solve the following equations, corr. to 2 d.p.~ 1. 3[5^(x+3)] = 4[7^(2x-1)] 2. [6^(x+1)]^2 = 2[3^(2x-1)] 3. (5/2)^(x-1)．(1/2)^x = 25/2 4. 2[6^(2-x)] - 3 =13 Given 8^x = 0.08^y = 1000. Without finding the values of x, y, find the value of 1/x - 1/y. 更新: ANS 1)2.84 2)-2.88 3)15.4 4)0.84 *last Q: 2/3
最佳解答:
1)3(5x+3)=4(72x-1) log3(5x+3)=log4(72x-1) log3+log5x+3=log4+log(72x-1) log4-log3=(x+3)log5-(2x-1)log7 log4/3=xlog5+3log5-2xlog7+log7 log4/3-3log5-log7=(log5-2log7)x x=2.84 (cor. to 2 d.p.) 2)(6x+1)2 = 2(32x-1) 62x+2=2(32x-1) log62x+2=log2(32x-1) (2x+2)log6=log2+log32x-1 2xlog6+2log6=log2+(2x-1)log3 2xlog6=log2-2log6+2xlog3-log3 x(2log6-2log3)=log2-2log6-log3 x=-2.88 (cor. to 2d.p.) 3)(5/2)x-1(1/2)x = 25/2 (5/2)x(2/5)(1/2)x = 25/2 [(5/2)(1/2)]x = 31.25 (5/4)x = 31.25 log (5/4)x = log 31.25 x log (5/4) = log 31.25 x = 15.43 ( cor. to 2 d.p. ) 4)2(62-x) - 3 =13 2(62-x)=16 62-x=8 (36)(6-x)=8 6-x=2/9 1/6x=2/9 6x=4.5 log6x=log4.5 xlog6=log4.5 x=0.84 ( cor. to 2d.p. ) 5)8x = 0.08y = 1000 So 8x = 1000 xlog8 = log1000 x = 3 / log 8 0.08y = 1000 ylog0.08= log1000 y=3/log0.08 = 3/[log(8/100)] = 3/(log8-2) Then, 1/x-1/y = 1/(3/log8)-1/[3/(log8-2)] =log8/3-(log8-2)/3 =(log8-log8+2)/3 =2/3
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