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10.6E: Mazoezi

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    Mazoezi hufanya kamili

    Zoezi\(\PageIndex{17}\) Solve Logarithmic Equations Using the Properties of Logarithms

    Katika mazoezi yafuatayo, tatua\(x\).

    1. \(\log _{4} 64=2 \log _{4} x\)
    2. \(\log 49=2 \log x\)
    3. \(3 \log _{3} x=\log _{3} 27\)
    4. \(3 \log _{6} x=\log _{6} 64\)
    5. \(\log _{5}(4 x-2)=\log _{5} 10\)
    6. \(\log _{3}\left(x^{2}+3\right)=\log _{3} 4 x\)
    7. \(\log _{3} x+\log _{3} x=2\)
    8. \(\log _{4} x+\log _{4} x=3\)
    9. \(\log _{2} x+\log _{2}(x-3)=2\)
    10. \(\log _{3} x+\log _{3}(x+6)=3\)
    11. \(\log x+\log (x+3)=1\)
    12. \(\log x+\log (x-15)=2\)
    13. \(\log (x+4)-\log (5 x+12)=-\log x\)
    14. \(\log (x-1)-\log (x+3)=\log \frac{1}{x}\)
    15. \(\log _{5}(x+3)+\log _{5}(x-6)=\log _{5} 10\)
    16. \(\log _{5}(x+1)+\log _{5}(x-5)=\log _{5} 7\)
    17. \(\log _{3}(2 x-1)=\log _{3}(x+3)+\log _{3} 3\)
    18. \(\log (5 x+1)=\log (x+3)+\log 2\)
    Jibu

    2. \(x=7\)

    4. \(x=4\)

    6. \(x=1, x=3\)

    8. \(x=8\)

    10. \(x=3\)

    12. \(x=20\)

    14. \(x=3\)

    16. \(x=6\)

    18. \(x=\frac{5}{3}\)

    Zoezi\(\PageIndex{18}\) Solve Exponential Equations Using Logarithms

    Katika mazoezi yafuatayo, tatua kila equation ya kielelezo. Pata jibu halisi na kisha uifanye karibu na maeneo matatu ya decimal.

    1. \(3^{x}=89\)
    2. \(2^{x}=74\)
    3. \(5^{x}=110\)
    4. \(4^{x}=112\)
    5. \(e^{x}=16\)
    6. \(e^{x}=8\)
    7. \(\left(\frac{1}{2}\right)^{x}=6\)
    8. \(\left(\frac{1}{3}\right)^{x}=8\)
    9. \(4 e^{x+1}=16\)
    10. \(3 e^{x+2}=9\)
    11. \(6 e^{2 x}=24\)
    12. \(2 e^{3 x}=32\)
    13. \(\frac{1}{4} e^{x}=3\)
    14. \(\frac{1}{3} e^{x}=2\)
    15. \(e^{x+1}+2=16\)
    16. \(e^{x-1}+4=12\)
    Jibu

    2. \(x=\frac{\log 74}{\log 2} \approx 6.209\)

    4. \(x=\frac{\log 112}{\log 4} \approx 3.404\)

    6. \(x=\ln 8 \approx 2.079\)

    8. \(x=\frac{\log 8}{\log \frac{1}{3}} \approx-1.893\)

    10. \(x=\ln 3-2 \approx-0.901\)

    12. \(x=\frac{\ln 16}{3} \approx 0.924\)

    14. \(x=\ln 6 \approx 1.792\)

    16. \(x=\ln 8+1 \approx 3.079\)

    Zoezi\(\PageIndex{19}\) Solve Exponential Equations Using Logarithms

    Katika mazoezi yafuatayo, tatua kila equation.

    1. \(3^{3 x+1}=81\)
    2. \(6^{4 x-17}=216\)
    3. \(\frac{e^{x^{2}}}{e^{14}}=e^{5 x}\)
    4. \(\frac{e^{x^{2}}}{e^{x}}=e^{20}\)
    5. \(\log _{a} 64=2\)
    6. \(\log _{a} 81=4\)
    7. \(\ln x=-8\)
    8. \(\ln x=9\)
    9. \(\log _{5}(3 x-8)=2\)
    10. \(\log _{4}(7 x+15)=3\)
    11. \(\ln e^{5 x}=30\)
    12. \(\ln e^{6 x}=18\)
    13. \(3 \log x=\log 125\)
    14. \(7 \log _{3} x=\log _{3} 128\)
    15. \(\log _{6} x+\log _{6}(x-5)=\log _{6} 24\)
    16. \(\log _{9} x+\log _{9}(x-4)=\log _{9} 12\)
    17. \(\log _{2}(x+2)-\log _{2}(2 x+9)=-\log _{2} x\)
    18. \(\log _{6}(x+1)-\log _{6}(4 x+10)=\log _{6} \frac{1}{x}\)
    Jibu

    2. \(x=5\)

    4. \(x=-4, x=5\)

    6. \(a=3\)

    8. \(x=e^{9}\)

    10. \(x=7\)

    12. \(x=3\)

    14. \(x=2\)

    16. \(x=6\)

    18. \(x=5\)

    Zoezi\(\PageIndex{20}\) Solve Exponential Equations Using Logarithms

    Katika mazoezi yafuatayo, tatua\(x\), kutoa jibu halisi pamoja na makadirio ya maeneo matatu ya decimal.

    1. \(6^{x}=91\)
    2. \(\left(\frac{1}{2}\right)^{x}=10\)
    3. \(7 e^{x-3}=35\)
    4. \(8 e^{x+5}=56\)
    Jibu

    2. \(x=\frac{\log 10}{\log \frac{1}{2}} \approx-3.322\)

    4. \(x=\ln 7-5 \approx-3.054\)

    Zoezi\(\PageIndex{21}\) Use Exponential Models in Applications

    Katika mazoezi yafuatayo, tatua.

    1. Sung Lee inawekeza $\(5,000\) akiwa na umri\(18\). Anatumaini uwekezaji utakuwa na thamani ya $\(10,000\) wakati anarudi\(25\). Ikiwa maslahi huchanganya kuendelea, takriban kiwango gani cha ukuaji atahitaji kufikia lengo lake? Je, hiyo ni matarajio ya kuridhisha?
    2. Alice inawekeza $ akiwa na\(15,000\) umri\(30\) kutoka kusaini ziada ya kazi yake mpya. Anatumaini uwekezaji utakuwa na thamani ya $\(30,000\) wakati anarudi\(40\). Ikiwa maslahi huchanganya kuendelea, takriban kiwango gani cha ukuaji atahitaji kufikia lengo lake?
    3. Coralee inawekeza $\(5,000\) katika akaunti ambayo misombo maslahi ya kila mwezi na chuma\(7\)%. Itachukua muda gani kwa pesa yake mara mbili?
    4. Simone inawekeza $\(8,000\) katika akaunti ambayo misombo riba robo mwaka na chuma\(5\)%. Itachukua muda gani kwa pesa zake mara mbili?
    5. Watafiti kumbukumbu kwamba baadhi ya bakteria idadi ya watu ulipungua kutoka\(100,000\)\(100\) kwa\(24\) saa. Kwa kiwango hiki cha kuoza, ni bakteria ngapi zitakavyokuwa na\(16\) masaa?
    6. Watafiti walirekodi kuwa idadi fulani ya bakteria ilishuka kutoka\(800,000\)\(500,000\) kwa\(6\) saa baada ya utawala wa dawa. Kwa kiwango hiki cha kuoza, ni bakteria ngapi zitakavyokuwa na\(24\) masaa?
    7. Virusi huchukua\(6\) siku ili mara mbili idadi yake ya awali\(\left(A=2 A_{0}\right)\). Itachukua muda gani ili idadi yake mara tatu?
    8. Bakteria huongeza idadi yake ya awali kwa\(24\) masaa\(\left(A=2 A_{0}\right)\). Idadi yake itakuwa kubwa kiasi gani katika\(72\) masaa?
    9. Kaboni-14 hutumiwa kwa dating ya kijiolojia ya kaboni. Nusu yake ya maisha ni\(5,730\) miaka. Ni kiasi gani cha sampuli ya\(100\) -gram ya Carbon-14 itaachwa kwa\(1000\) miaka?
    10. Mionzi technetium-99m mara nyingi hutumika katika dawa za uchunguzi kwani ina nusu ya maisha mafupi kiasi lakini hudumu kwa muda mrefu wa kutosha kupata upimaji unaohitajika kufanyika kwa mgonjwa. Ikiwa nusu yake ya maisha ni\(6\) masaa, ni kiasi gani cha nyenzo za mionzi huunda sindano ya\(0.5\) ml itakuwa katika mwili kwa\(24\) masaa?
    Jibu

    2. \(6.9\)%

    4. \(13.9\)miaka

    6. \(122,070\)bakteria

    8. \(8\)mara kubwa kama idadi ya awali

    10. \(0.03\)mL

    Zoezi\(\PageIndex{22}\) Writing Exercises
    1. Eleza njia unayotumia kutatua milinganyo haya:\(3^{x+1}=81\),\(3^{x+1}=75\). Je, njia yako inahitaji logarithms kwa equations zote mbili? Kwa nini au kwa nini?
    2. ni tofauti kati ya equation kwa ajili ya ukuaji kielelezo dhidi equation kwa ajili ya kuoza kielelezo nini?
    Jibu

    2. Majibu yatatofautiana.

    Self Check

    Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

    Jedwali hili lina safu nne na nguzo nne. mstari wa kwanza, ambayo hutumika kama header, anasoma mimi canâ € |, Kwa ujasiri, Kwa msaada fulani, na Noâ €” Mimi donâ €™ t kupata hiyo. Safu ya kwanza chini ya mstari wa kichwa inasoma kutatua equations ya logarithmic kwa kutumia mali ya logarithms, kutatua equations kielelezo kwa kutumia logarithms, na kutumia mifano ya kielelezo katika programu. Wengine wa seli ni tupu.
    Kielelezo 10.5.1

    b Baada ya kuangalia orodha, unafikiri umeandaliwa vizuri kwa sehemu inayofuata? Kwa nini au kwa nini?