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  إجابات كتاب التمارين

إجابات كتاب التمارين

قوانين اللوغاريتمات

إذا كان: loga 7 ≈ 0.936 ، وكان: loga 3 ≈ 0.528 ، فأجد كلاً ممّا يأتي:

(1) begin mathsize 20px style log subscript a invisible function application space 3 over 7 end style

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(2) begin mathsize 20px style log subscript a invisible function application space 21 end style

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(3) begin mathsize 20px style fraction numerator log subscript a invisible function application space 3 over denominator log subscript a invisible function application space 7 end fraction end style

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(4) begin mathsize 20px style log subscript a invisible function application space 1 over 7 end style

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(5) begin mathsize 20px style log subscript a invisible function application space 441 end style

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(6) begin mathsize 20px style log subscript a invisible function application space 49 over 27 end style

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(7) begin mathsize 20px style log subscript a invisible function application space left parenthesis 7 a squared right parenthesis end style

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(8) begin mathsize 20px style log subscript a invisible function application space fourth root of 81 end style

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(9) begin mathsize 20px style left parenthesis log subscript a invisible function application space 3 right parenthesis left parenthesis log subscript a invisible function application space 7 right parenthesis end style

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أكتب كل مقدار لوغاريتمي ممّا يأتي بالصورة المطولة، علماً بأنّ المتغيرات جميعها تُمثّل أعداداً حقيقية موجبة:

(10) begin mathsize 20px style log subscript a invisible function application space x to the power of 7 end style

begin mathsize 20px style l o g subscript a invisible function application space x to the power of 7 equals 7 l o g subscript a invisible function application space x end style

(11) begin mathsize 20px style log subscript a invisible function application left parenthesis fraction numerator a c over denominator b end fraction right parenthesis end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell l o g subscript a invisible function application left parenthesis fraction numerator a c over denominator b end fraction right parenthesis end cell cell equals l o g subscript a invisible function application a c minus l o g subscript a invisible function application b end cell row blank cell equals l o g subscript a invisible function application a plus l o g subscript a invisible function application c minus l o g subscript a invisible function application b end cell row blank cell equals 1 plus l o g subscript a invisible function application c minus l o g subscript a invisible function application b end cell end table end style

(12) begin mathsize 20px style log subscript a invisible function application left parenthesis square root of x right parenthesis end style

begin mathsize 20px style l o g subscript a invisible function application left parenthesis square root of x right parenthesis equals l o g subscript a invisible function application x to the power of 1 half end exponent equals 1 half l o g subscript a invisible function application x end style

(13) begin mathsize 20px style log subscript a invisible function application left parenthesis fraction numerator square root of x y end root over denominator z end fraction right parenthesis end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell l o g subscript a invisible function application left parenthesis fraction numerator square root of x y end root over denominator z end fraction right parenthesis end cell cell equals l o g subscript a invisible function application square root of x y end root minus l o g subscript a invisible function application z end cell row blank cell equals l o g subscript a invisible function application left parenthesis x y right parenthesis to the power of 1 half end exponent minus l o g subscript a invisible function application z end cell row equals cell l o g subscript a invisible function application x to the power of 1 half end exponent y to the power of 1 half end exponent minus l o g subscript a invisible function application z end cell row equals cell l o g subscript a invisible function application x to the power of 1 half end exponent plus l o g subscript a invisible function application y to the power of 1 half end exponent minus l o g subscript a invisible function application z end cell row blank cell equals 1 half l o g subscript a invisible function application x plus 1 half l o g subscript a invisible function application y minus l o g subscript a invisible function application z end cell end table end style 

(14) begin mathsize 20px style log subscript a invisible function application fraction numerator 1 over denominator x cubed y to the power of 4 end fraction end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell l o g subscript a invisible function application fraction numerator 1 over denominator x cubed y to the power of 4 end fraction end cell cell equals l o g subscript a invisible function application 1 minus l o g subscript a invisible function application x cubed y to the power of 4 end cell row blank cell equals l o g subscript a invisible function application 1 minus left parenthesis l o g subscript a invisible function application x cubed plus l o g subscript a invisible function application y to the power of 4 right parenthesis end cell row blank cell equals 0 minus left parenthesis 3 l o g subscript a invisible function application x plus 4 l o g subscript a invisible function application y right parenthesis end cell row blank cell equals negative 3 l o g subscript a invisible function application x minus 4 l o g subscript a invisible function application y end cell end table end style

(15) begin mathsize 20px style log subscript a invisible function application root index 7 of 128 x to the power of 7 end root end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell l o g subscript a invisible function application root index 7 of 128 x to the power of 7 end root end cell cell equals l o g subscript a invisible function application root index 7 of 128 cross times root index 7 of x to the power of 7 end root end cell row blank cell equals l o g subscript a invisible function application 2 x end cell row blank cell equals l o g subscript a invisible function application 2 plus l o g subscript a invisible function application x end cell end table end style

(16) begin mathsize 20px style log subscript a invisible function application left parenthesis x to the power of negative 1 end exponent y squared right parenthesis to the power of 4 over left parenthesis x to the power of 5 y to the power of negative 2 end exponent right parenthesis cubed end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell l o g subscript a invisible function application left parenthesis x to the power of negative 1 end exponent y squared right parenthesis to the power of 4 over left parenthesis x to the power of 5 y to the power of negative 2 end exponent right parenthesis cubed end cell cell equals l o g subscript a invisible function application fraction numerator x to the power of negative 4 end exponent y to the power of 8 over denominator x to the power of 15 y to the power of negative 6 end exponent end fraction end cell row blank cell equals l o g subscript a invisible function application x to the power of negative 19 end exponent y to the power of 14 end cell row blank cell equals l o g subscript a invisible function application x to the power of negative 19 end exponent plus l o g subscript a invisible function application y to the power of 14 end cell row blank cell equals negative 19 l o g subscript a invisible function application x plus 14 l o g subscript a invisible function application y end cell end table end style

(17) begin mathsize 20px style log subscript a invisible function application square root of fraction numerator x squared y cubed over denominator z cubed end fraction end root end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell l o g subscript a invisible function application square root of fraction numerator x squared y cubed over denominator z cubed end fraction end root end cell cell equals l o g subscript a invisible function application fraction numerator square root of x squared end root square root of y cubed end root over denominator square root of z cubed end root end fraction end cell row blank cell equals l o g subscript a invisible function application fraction numerator x y to the power of 3 over 2 end exponent over denominator z to the power of 3 over 2 end exponent end fraction end cell row blank cell equals l o g subscript a invisible function application x y to the power of 3 over 2 end exponent minus l o g subscript a invisible function application z to the power of 3 over 2 end exponent end cell row blank cell equals l o g subscript a invisible function application x plus l o g subscript a invisible function application y to the power of 3 over 2 end exponent minus l o g subscript a invisible function application z to the power of 3 over 2 end exponent end cell row blank cell equals l o g subscript a invisible function application x plus 3 over 2 l o g subscript a invisible function application y minus 3 over 2 l o g subscript a invisible function application z end cell end table end style  

(18) begin mathsize 20px style log subscript a invisible function application left parenthesis x minus y plus z right parenthesis to the power of 9 comma space y minus x less than z end style

begin mathsize 20px style l o g subscript a invisible function application left parenthesis x minus y plus z right parenthesis to the power of 9 equals 9 l o g subscript a invisible function application left parenthesis x minus y plus z right parenthesis end style  

 

أكتب كل مقدار لوغاريتمي ممّا يأتي بالصورة المختصرة، علماً بأنّ المتغيرات جميعها تُمثّل أعداداً حقيقية موجبة:

(19) begin mathsize 20px style log subscript a invisible function application space x minus log subscript a invisible function application space y end style

begin mathsize 20px style l o g subscript a invisible function application x minus l o g subscript a invisible function application y equals l o g subscript a invisible function application x over y end style

(20) begin mathsize 20px style log subscript b invisible function application left parenthesis b minus 1 right parenthesis plus 2 log subscript b invisible function application b comma space b greater than 1 end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell l o g subscript b invisible function application left parenthesis b minus 1 right parenthesis plus 2 l o g subscript b invisible function application b end cell cell equals l o g subscript b invisible function application left parenthesis b minus 1 right parenthesis plus l o g subscript b invisible function application b squared end cell row blank cell equals l o g subscript b invisible function application fraction numerator b minus 1 over denominator b squared end fraction end cell end table end style

(21) begin mathsize 20px style log subscript a invisible function application square root of x minus log subscript a invisible function application fraction numerator 1 over denominator square root of x end fraction end style

begin mathsize 20px style l o g subscript a invisible function application square root of x minus l o g subscript a invisible function application fraction numerator 1 over denominator square root of x end fraction equals l o g subscript a invisible function application fraction numerator square root of x over denominator fraction numerator 1 over denominator square root of x end fraction end fraction equals l o g invisible function application x end style

(22) begin mathsize 20px style log subscript a invisible function application left parenthesis x squared minus 25 right parenthesis minus log subscript a invisible function application left parenthesis x plus 5 right parenthesis comma space x greater than 5 to the power of blank end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell l o g subscript a invisible function application left parenthesis x squared minus 25 right parenthesis minus l o g subscript a invisible function application left parenthesis x plus 5 right parenthesis end cell cell equals l o g subscript a invisible function application fraction numerator left parenthesis x squared minus 25 right parenthesis over denominator left parenthesis x plus 5 right parenthesis end fraction end cell row blank cell equals l o g subscript a invisible function application fraction numerator left parenthesis x plus 5 right parenthesis left parenthesis x minus 5 right parenthesis over denominator left parenthesis x plus 5 right parenthesis end fraction end cell row blank cell equals l o g subscript a invisible function application left parenthesis x minus 5 right parenthesis end cell end table end style

(23) begin mathsize 20px style 3 log subscript b invisible function application 1 minus log subscript b invisible function application b end style

begin mathsize 20px style 3 l o g subscript b invisible function application 1 minus l o g subscript b invisible function application b equals 3 left parenthesis 0 right parenthesis minus 1 equals negative 1 end style

(24) begin mathsize 20px style 8 log subscript b invisible function application x plus 4 log subscript b invisible function application y minus 1 half log subscript b invisible function application z end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 8 l o g subscript b invisible function application x plus 4 l o g subscript b invisible function application y minus 1 half l o g subscript b invisible function application z end cell cell equals l o g subscript b invisible function application x to the power of 8 plus l o g subscript b invisible function application y to the power of 4 minus l o g subscript b invisible function application z to the power of 1 half end exponent end cell row blank cell equals l o g subscript b invisible function application x to the power of 8 y to the power of 4 minus l o g subscript b invisible function application z to the power of 1 half end exponent end cell row blank cell equals l o g subscript b invisible function application fraction numerator x to the power of 8 y to the power of 4 over denominator z to the power of 1 half end exponent end fraction equals l o g subscript b invisible function application fraction numerator x to the power of 8 y to the power of 4 over denominator square root of z end fraction end cell end table end style

 

(25) إيرادات: يمثل الاقتران: T(a) = 10 + 20 log6 (a + 1) مبيعات شركة (بآلاف الدنانير) من منتج جديد، حيث a المبلغ (بآلاف الدنانير) الذي تنفقه الشركة على إعلانات المنتج، و a greater or equal than 0 . وتعني القيمة: T(1) ≈ 17.7 أنّ إنفاق JD 1000 على الإعلانات يحقق إيرادات قيمتها JD 177000 من بيع المنتج. أجد قيمة إيرادات الشركة بعد إنفاقها مبلغ 11 ألف دينار على الإعلانات، علماً بأنّ log6 2 ≈ 0.3869 .

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قيمة إيرادات الشركة بعد إنفاقها مبلغ JD 11000 على الإعلانات هو JD 37738 .

إعداد : شبكة منهاجي التعليمية

10 / 07 / 2023

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