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10.4E: Mazoezi

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    Sehemu ya 10.3 Mazoezi

    Mazoezi hufanya kamili

    Zoezi\(\PageIndex{21}\) Convert Between Exponential and Logarithmic Form

    Katika mazoezi yafuatayo, kubadilisha kutoka kwa kielelezo hadi fomu ya logarithmic.

    1. \(4^{2}=16\)
    2. \(2^{5}=32\)
    3. \(3^{3}=27\)
    4. \(5^{3}=125\)
    5. \(10^{3}=1000\)
    6. \(10^{-2}=\frac{1}{100}\)
    7. \(x^{\frac{1}{2}}=\sqrt{3}\)
    8. \(x^{\frac{1}{3}}=\sqrt[3]{6}\)
    9. \(32^{x}=\sqrt[4]{32}\)
    10. \(17^{x}=\sqrt[5]{17}\)
    11. \(\left(\frac{1}{4}\right)^{2}=\frac{1}{16}\)
    12. \(\left(\frac{1}{3}\right)^{4}=\frac{1}{81}\)
    13. \(3^{-2}=\frac{1}{9}\)
    14. \(4^{-3}=\frac{1}{64}\)
    15. \(e^{x}=6\)
    16. \(e^{3}=x\)
    Jibu

    2. \(\log _{2} 32=5\)

    4. \(\log _{5} 125=3\)

    6. \(\log \frac{1}{100}=-2\)

    8. \(\log _{x} \sqrt[3]{6}=\frac{1}{3}\)

    10. \(\log _{17} \sqrt[5]{17}=x\)

    12. \(\log _{\frac{1}{3}} \frac{1}{81}=4\)

    14. \(\log _{4} \frac{1}{64}=-3\)

    16. \(\ln x=3\)

    Zoezi\(\PageIndex{22}\) Convert Between Exponential and Logarithmic Form

    Katika mazoezi yafuatayo, kubadilisha kila equation ya logarithmic kwa fomu ya kielelezo.

    1. \(3=\log _{4} 64\)
    2. \(6=\log _{2} 64\)
    3. \(4=\log _{x} 81\)
    4. \(5=\log _{x} 32\)
    5. \(0=\log _{12} 1\)
    6. \(0=\log _{7} 1\)
    7. \(1=\log _{3} 3\)
    8. \(1=\log _{9} 9\)
    9. \(-4=\log _{10} \frac{1}{10,000}\)
    10. \(3=\log _{10} 1,000\)
    11. \(5=\log _{e} x\)
    12. \(x=\log _{e} 43\)
    Jibu

    2. \(64=2^{6}\)

    4. \(32=x^{5}\)

    6. \(1=7^{0}\)

    8. \(9=9^{1}\)

    10. \(1,000=10^{3}\)

    12. \(43=e^{x}\)

    Zoezi\(\PageIndex{23}\) Evaluate Logarithmic Functions

    Katika mazoezi yafuatayo, pata thamani ya\(x\) kila equation ya logarithmic.

    1. \(\log _{x} 49=2\)
    2. \(\log _{x} 121=2\)
    3. \(\log _{x} 27=3\)
    4. \(\log _{x} 64=3\)
    5. \(\log _{3} x=4\)
    6. \(\log _{5} x=3\)
    7. \(\log _{2} x=-6\)
    8. \(\log _{3} x=-5\)
    9. \(\log _{\frac{1}{4}} \frac{1}{16}=x\)
    10. \(\log _{\frac{1}{3}} \frac{1}{9}=x\)
    11. \(\log _{\frac{1}{4}} 64=x\)
    12. \(\log _{\frac{1}{9}} 81=x\)
    Jibu

    2. \(x=11\)

    4. \(x=4\)

    6. \(x=125\)

    8. \(x=\frac{1}{243}\)

    10. \(x=2\)

    12. \(x=-2\)

    Zoezi\(\PageIndex{24}\) Evaluate Logarithmic Functions

    Katika mazoezi yafuatayo, pata thamani halisi ya kila logarithm bila kutumia calculator.

    1. \(\log _{7} 49\)
    2. \(\log _{6} 36\)
    3. \(\log _{4} 1\)
    4. \(\log _{5} 1\)
    5. \(\log _{16} 4\)
    6. \(\log _{27} 3\)
    7. \(\log _{\frac{1}{2}} 2\)
    8. \(\log _{\frac{1}{2}} 4\)
    9. \(\log _{2} \frac{1}{16}\)
    10. \(\log _{3} \frac{1}{27}\)
    11. \(\log _{4} \frac{1}{16}\)
    12. \(\log _{9} \frac{1}{81}\)
    Jibu

    2. \(2\)

    4. \(0\)

    6. \(\frac{1}{3}\)

    8. \(-2\)

    10. \(-3\)

    12. \(-2\)

    Zoezi\(\PageIndex{25}\) Graph Logarithmic Functions

    Katika mazoezi yafuatayo, grafu kila kazi ya logarithmic.

    1. \(y=\log _{2} x\)
    2. \(y=\log _{4} x\)
    3. \(y=\log _{6} x\)
    4. \(y=\log _{7} x\)
    5. \(y=\log _{1.5} x\)
    6. \(y=\log _{2.5} x\)
    7. \(y=\log _{\frac{1}{3}} x\)
    8. \(y=\log _{\frac{1}{5}} x\)
    9. \(y=\log _{0.4} x\)
    10. \(y=\log _{0.6} x\)
    Jibu

    2.

    Takwimu hii inaonyesha Curve logarithmic kupitia pointi (1 juu ya 4, hasi 1), (1, 0), na (4, 1).
    Kielelezo 10.3.19

    4.

    Takwimu hii inaonyesha kwamba Curve logarithmic kupitia pointi (1 juu ya 7, hasi 1), (1, 0), na (7, 1).
    Kielelezo 10.3.20

    6.

    Takwimu hii inaonyesha safu ya logarithmic inayopitia pointi (2 juu ya 5, hasi 1), (1, 0), na (2.5, 1).
    Kielelezo 10.3.21

    8.

    Takwimu hii inaonyesha Curve logarithmic kupitia pointi (1 juu ya 5, 1), (1, 0), na (5, hasi 1).
    Kielelezo 10.3.22

    10.

    Takwimu hii inaonyesha Curve logarithmic kupitia pointi (3 juu ya 5, 1), (1, 0), na (5 juu ya 3, hasi 1).
    Kielelezo 10.3.23
    Zoezi\(\PageIndex{26}\) Solve Logarithmic Equations

    Katika mazoezi yafuatayo, tatua kila equation ya logarithmic.

    1. \(\log _{a} 16=2\)
    2. \(\log _{a} 81=2\)
    3. \(\log _{a} 8=3\)
    4. \(\log _{a} 27=3\)
    5. \(\log _{a} 32=2\)
    6. \(\log _{a} 24=3\)
    7. \(\ln x=5\)
    8. \(\ln x=4\)
    9. \(\log _{2}(5 x+1)=4\)
    10. \(\log _{2}(6 x+2)=5\)
    11. \(\log _{3}(4 x-3)=2\)
    12. \(\log _{3}(5 x-4)=4\)
    13. \(\log _{4}(5 x+6)=3\)
    14. \(\log _{4}(3 x-2)=2\)
    15. \(\ln e^{4 x}=8\)
    16. \(\ln e^{2 x}=6\)
    17. \(\log x^{2}=2\)
    18. \(\log \left(x^{2}-25\right)=2\)
    19. \(\log _{2}\left(x^{2}-4\right)=5\)
    20. \(\log _{3}\left(x^{2}+2\right)=3\)
    Jibu

    2. \(a=9\)

    4. \(a=3\)

    6. \(a=\sqrt[3]{24}\)

    8. \(x=e^{4}\)

    10. \(x=5\)

    12. \(x=17\)

    14. \(x=6\)

    16. \(x=3\)

    18. \(x=-5 \sqrt{5}, x=5 \sqrt{5}\)

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

    Zoezi\(\PageIndex{27}\) Use Logarithmic Models in Applications

    Katika mazoezi yafuatayo, tumia mfano wa logarithmic kutatua.

    1. Je, ni kiwango cha decibel cha mazungumzo ya kawaida na\(10^{−6}\) watts kali kwa inchi ya mraba?
    2. Je! Ni kiwango gani cha decibel cha whisper na\(10^{−10}\) watts kali kwa inchi ya mraba?
    3. Je, ni kiwango cha decibel cha kelele kutoka kwa pikipiki na\(10^{−2}\) watts kali kwa inchi ya mraba?
    4. Je! Ni kiwango gani cha decibel cha sauti ya taka ya taka na\(10^{−2}\) watts kali kwa inchi ya mraba?
    5. Mwaka 2014, Chile ilipata tetemeko kubwa la ardhi na ukubwa wa\(8.2\) kiwango cha Richter. Mwaka 2010, Haiti pia ilipata tetemeko kubwa la ardhi ambalo\(7.0\) lilipimwa kwa kiwango cha Richter. Kulinganisha intensities ya tetemeko la ardhi mbili.
    6. Eneo la Los Angeles linapata matetemeko mengi. Mwaka 1994, tetemeko la ardhi la Northridge lilipima ukubwa wa\(6.7\) kiwango cha Richter. Mwaka 2014, Los Angeles pia ilipata tetemeko la ardhi ambalo\(5.1\) lilipimwa kwa kiwango cha Richter. Kulinganisha intensities ya tetemeko la ardhi mbili.
    Jibu

    2. Whisper ina kiwango cha decibel cha\(20\) dB.

    4. Sauti ya ovyo ya takataka ina kiwango cha decibel cha\(100\) dB.

    6. Ukubwa wa tetemeko la ardhi la Northridge la 1994 katika eneo la Los Angeles lilikuwa karibu\(40\) mara ukubwa wa tetemeko la ardhi la 2014.

    Zoezi\(\PageIndex{28}\) Writing Exercises
    1. Eleza jinsi ya kubadilisha equation kutoka fomu ya logarithmic kwa fomu ya kielelezo.
    2. Eleza tofauti kati ya logarithms ya kawaida na logarithms ya asili.
    3. Eleza kwa nini\(\log _{a} a^{x}=x\).
    4. Eleza jinsi ya kupata\(\log _{7} 32\) kwenye calculator yako.
    Jibu

    2. Majibu inaweza kutofautiana

    4. Majibu inaweza kutofautiana

    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 tano. 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 Badilisha kati ya fomu ya kielelezo na logarithmic, tathmini kazi za logarithmic, kazi za logarithmic, kutatua equations ya logarithmic, na kutumia mifano ya logarithmic katika programu. Wengine wa seli ni tupu.
    Kielelezo 10.3.24

    b Baada ya kuchunguza orodha hii, utafanya nini ili uwe na ujasiri kwa malengo yote?