發問時間: 科學數學 · 6 年前

指數與對數計算詳解

*請詳解

(1)2^127-1是一個質數,此質數是幾位數? 答:39

(2)(1/5)^50在小數點第幾位才出現不為0的數字? 答:7

(3)設a=log5(底數)2,b=log2(底數)3,則log18=? 答:a+2ab/a+1

(4)logx首數與log345的首數相同,logx的尾數與log12.5的尾數相同,則x=? 答:125

(5)已知A為整數,且2.2>logA>2.1,則A^3為幾位數? 答:7

(6)設x為實數,且6^X-3^X+2^X-1=0,則6^X=? 答:1

(7)若log3(底數)x+log3(底數)y=2,則1/x+1/y之最小值為何? 答:2/3

(8)設n為自然數,且x=3^n+3^n+3^n,則下列何者等於x^2? 答:3^(2n+2)

(9)若log2(底數)(x^2-x-1)之值恆為正,則x之範圍為何? 答:x<1或x>2

請解釋為何不是1<x<2

(10)哪個函數f(x)具有"對於任意兩正實數a及b,f(ab)=f(a)+f(b)均為成立的性質"?

答:f(x)=logx

(11)設x、y為正實數,若2log(x-2y)=logx+logy,則x/y之值為何? 答:4

*請詳解

2 個解答

評分
  • 6 年前
    最佳解答

    (1)2^127-1是一個質數,此質數是幾位數?

    Sol

    a=2^127-1

    log(2^127-1)~log2^127=127log2=38.2257

    log2=0.3010 =>2是1位數

    log20=1.3010=1+0.3010 =>20是2位數

    log200=2.3010=2+0.3010 =>200是3位數

    So

    2^127-1是39位數

    (2)(1/5)^50在小數點第幾位才出現不為0的數字?

    Sol

    a=(1/5)^50

    loga=50log(1/5)=50(log2-log10)=50*(0.3010-1)=-34.95=-35+0.05

    log0.011=-2+log1.1 =>在小數點第2位才出現不為0的數字

    log0.0011=-3+log1.1 =>在小數點第3位才出現不為0的數字

    So

    (1/5)^50在小數點第35位才出現不為0的數字

    (3)設a=log5_2,b=log2_3,則log18=?

    Sol

    a=log2/log5=log2/(1-log2)

    a-alog2=log2

    a=(1+a)log2

    log2=a/(1+a)

    b=log3/log2

    log3=blog2=ab(1+a)

    log18=log2+2log3

    =a/(1+a)+2ab(1+a)

    =(a+2ab)/(a+1)

    (4)logx首數與log345的首數相同,logx的尾數與log12.5的尾數相同,則x=?

    Sol

    log345=log100+log3.45=2+log3.45

    log12.5=log10+log1.25=1+log1.25

    logx=2+log1.25=log125

    x=125

    (5)已知a為整數,且2.2>loga>2.1,則a^3為幾位數?

    Sol

    2.2>loga>2.1

    10^2.2>a>10^2.1

    10^6.6>a^3<10^6.3

    6+1=7

    a^3為7位數?

    (6)設x為實數,且6^x-3^x+2^x-1=0,則6^x=?

    Sol

    6^x-3^x+2^x-1=0

    (6^x-3^x)+(2^x-1)=0

    3^x(2^x-1)+(2^x-1)=0

    (3^x+1)(2^x-1)=0

    2^x-1=0

    2^x=1

    x=0

    6^x=6^0=1

    (7)若log3_x+log3_y=2,則1/x+1/y之最小值為何?

    Sol

    x>0,y>0

    log3_(xy)=2

    xy=9

    (1/x+1/y)/2>=√[(1/x)*(1/y)]

    (1/x+1/y)>=2√[(1/x)*(1/y)]=2√(1/9)

    (1/x+1/y)>=2/3

    (8)設n為自然數,且x=3^n+3^n+3^n,則下列何者等於x^2?

    Sol

    x=3*3^n=3^(n+1)

    x^2=3^(2n+2)

    (9)若log2_(x^2-x-1)之值恆為正,則x之範圍為何?

    Sol

    log2_(x^2-x-1)>0

    x^2-x-1>1

    x^2-x-2>0

    (x-2)(x+1)>0

    x-2>0 or x+1<0

    x>2 orx<-1

    (11)設x、y為正實數,若2log(x-2y)=logx+logy,則x/y之值為何?

    Sol

    2log(x-2y)=logx+logy

    (x-2y)^2=xy

    x^2-4xy+4y^2=xy

    x^2-5xy+4y^2=0

    (x-y)(x-4y)=0

    (1) x=y

    X-2y=-x<0(不合)

    (2) x=4y

    x/y\=4

  • 阿番
    Lv 7
    6 年前

    11.

    x-2y>0

    ==>x>2y

    2log(x-2y)=logx+logy

    ==>log(x-2y)^2=logxy

    ==>(x-2y)^2=xy

    ==>(x-4y)(x-y)=0

    ==>x=4y或(x=y不合因x>2y)

    x/y=4y/y=4

    2015-01-11 14:18:43 補充:

    (3)設a=log5(底數)2,b=log2(底數)3,則log18=? 答:a+2ab/a+1

    a=log5(底數)2=log2/log5

    ==>alog5=log2

    ==>a(1-log2)=log2

    ==>log2=a/(a+1)

    b=log2(底數)3=log3/log2

    ==>blog2=log3

    log18=log3^2+log2=2log3+log2=2blog2+log2=(2b+1)log2=a(2b+1)/(a+1)

    =(a+2ab)/(a+1)

    2015-01-11 14:19:14 補充:

    (7)若log3(底數)x+log3(底數)y=2,則1/x+1/y之最小值為何?? 答:2/3

    log3(底數)x+log3(底數)y=2

    ==>log3_xy=log3_3^2

    ==>xy=9

    1/x+1/y>=2((1/x)*(1/y))^(1/2)=(1/9)^(1/2) 算幾不等式

    ==>1/x+1/y>=2/3

    其最小值為2/3

    2015-01-11 14:39:51 補充:

    (6)設x為實數,且6^X-3^X+2^X-1=0,則6^X=? 答:1

    令2^X=u,3^X=v

    6^X-3^X+2^X-1=0

    ==>(3^X)(2^X)-3^X+2^X-1=0

    ==>uv-v+u-1=0

    ==>u(v+1)-(v+1)=0

    ==>(u-1)(v+1)=0

    ==>u=1,(v=-1不合因真數要大於0)

    2^X=1 ==>X=0

    6^X=6^0=1

    2015-01-11 14:43:47 補充:

    (8)設n為自然數,且x=3^n+3^n+3^n,則下列何者等於x^2? 答:3^(2n+2)

    x=3^n+3^n+3^n

    ==>x=3*(3^n)=3^(n+1)

    ==>x^2=(3^(n+1))^2=3^(2n+2)

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