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10.5E: Mazoezi

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    Mazoezi hufanya kamili

    Zoezi\(\PageIndex{21}\) Use the Properties of Logarithms

    Katika mazoezi yafuatayo, tumia mali ya logarithms kutathmini.

      1. \(\log _{4} 1\)
      2. \(\log _{8} 8\)
      1. \(\log _{12} 1\)
      2. \(\ln e\)
      1. \(3^{\log _{3} 6}\)
      2. \(\log _{2} 2^{7}\)
      1. \(5^{\log _{5} 10}\)
      2. \(\log _{4} 4^{10}\)
      1. \(8^{\log _{8} 7}\)
      2. \(\log _{6} 6^{-2}\)
      1. \(6^{\log _{6} 15}\)
      2. \(\log _{8} 8^{-4}\)
      1. \(10^{\log \sqrt{5}}\)
      2. \(\log 10^{-2}\)
      1. \(10^{\log \sqrt{3}}\)
      2. \(\log 10^{-1}\)
      1. \(e^{\ln 4}\)
      2. \(\ln e^{2}\)
      1. \(e^{\ln 3}\)
      2. \(\ln e^{7}\)
    Jibu

    2.

    1. \(0\)
    2. \(1\)

    4.

    1. \(10\)
    2. \(10\)

    6.

    1. \(15\)
    2. \(-4\)

    8.

    1. \(\sqrt{3}\)
    2. \(-1\)

    10.

    1. \(3\)
    2. \(7\)
    Zoezi\(\PageIndex{22}\) Use the Properties of Logarithms

    Katika mazoezi yafuatayo, tumia Mali ya Bidhaa ya Logarithms kuandika kila logarithm kama jumla ya logarithms. Kurahisisha kama inawezekana.

    1. \(\log _{4} 6 x\)
    2. \(\log _{5} 8 y\)
    3. \(\log _{2} 32 x y\)
    4. \(\log _{3} 81 x y\)
    5. \(\log 100 x\)
    6. \(\log 1000 y\)
    Jibu

    2. \(\log _{5} 8+\log _{5} y\)

    4. \(4+\log _{3} x+\log _{3} y\)

    6. \(3+\log y\)

    Zoezi\(\PageIndex{23}\) Use the Properties of Logarithms

    Katika mazoezi yafuatayo, tumia Mali ya Quotient ya Logarithms kuandika kila logarithm kama jumla ya logarithms. Kurahisisha kama inawezekana.

    1. \(\log _{3} \frac{3}{8}\)
    2. \(\log _{6} \frac{5}{6}\)
    3. \(\log _{4} \frac{16}{y}\)
    4. \(\log _{5} \frac{125}{x}\)
    5. \(\log \frac{x}{10}\)
    6. \(\log \frac{10,000}{y}\)
    7. \(\ln \frac{e^{3}}{3}\)
    8. \(\ln \frac{e^{4}}{16}\)
    Jibu

    2. \(\log _{6} 5-1\)

    4. \(3-\log _{5} x\)

    6. \(4-\log y\)

    8. \(4-\ln 16\)

    Zoezi\(\PageIndex{24}\) Use the Properties of Logarithms

    Katika mazoezi yafuatayo, tumia Mali ya Nguvu ya Logarithms kupanua kila mmoja. Kurahisisha kama inawezekana.

    1. \(\log _{3} x^{2}\)
    2. \(\log _{2} x^{5}\)
    3. \(\log x^{-2}\)
    4. \(\log x^{-3}\)
    5. \(\log _{4} \sqrt{x}\)
    6. \(\log _{5} \sqrt[3]{x}\)
    7. \(\ln x^{\sqrt{3}}\)
    8. \(\ln x^{\sqrt[3]{4}}\)
    Jibu

    2. \(5\log _{2} x\)

    4. \(-3 \log x\)

    6. \(\frac{1}{3} \log _{5} x\)

    8. \(\sqrt[3]{4} \ln x\)

    Zoezi\(\PageIndex{25}\) Use the Properties of Logarithms

    Katika mazoezi yafuatayo, tumia Mali ya Logarithms kupanua logarithm. Kurahisisha kama inawezekana.

    1. \(\log _{5}\left(4 x^{6} y^{4}\right)\)
    2. \(\log _{2}\left(3 x^{5} y^{3}\right)\)
    3. \(\log _{3}\left(\sqrt{2} x^{2}\right)\)
    4. \(\log _{5}\left(\sqrt[4]{21} y^{3}\right)\)
    5. \(\log _{3} \frac{x y^{2}}{z^{2}}\)
    6. \(\log _{5} \frac{4 a b^{3} c^{4}}{d^{2}}\)
    7. \(\log _{4} \frac{\sqrt{x}}{16 y^{4}}\)
    8. \(\log _{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}\)
    9. \(\log _{2} \frac{\sqrt{2 x+y^{2}}}{z^{2}}\)
    10. \(\log _{3} \frac{\sqrt{3 x+2 y^{2}}}{5 z^{2}}\)
    11. \(\log _{2} \sqrt[4]{\frac{5 x^{3}}{2 y^{2} z^{4}}}\)
    12. \(\log _{5} \sqrt[3]{\frac{3 x^{2}}{4 y^{3} z}}\)
    Jibu

    2. \(\log _{2} 3+5 \log _{2} x+3 \log _{2} y\)

    4. \(\frac{1}{4} \log _{5} 21+3 \log _{5} y\)

    6. \(\begin{array}{l}{\log _{5} 4+\log _{5} a+3 \log _{5} b} {+4 \log _{5} c-2 \log _{5} d}\end{array}\)

    8. \(\frac{2}{3} \log _{3} x-3-4 \log _{3} y\)

    10. \(\frac{1}{2} \log _{3}\left(3 x+2 y^{2}\right)-\log _{3} 5-2 \log _{3} z\)

    12. \(\begin{array}{l}{\frac{1}{3}\left(\log _{5} 3+2 \log _{5} x-\log _{5} 4\right.} {-3 \log _{5} y-\log _{5} z )}\end{array}\)

    Zoezi\(\PageIndex{26}\) Use the Properties of Logarithms

    Katika mazoezi yafuatayo, tumia Mali ya Logarithms ili kuimarisha logarithm. Kurahisisha kama inawezekana.

    1. \(\log _{6} 4+\log _{6} 9\)
    2. \(\log 4+\log 25\)
    3. \(\log _{2} 80-\log _{2} 5\)
    4. \(\log _{3} 36-\log _{3} 4\)
    5. \(\log _{3} 4+\log _{3}(x+1)\)
    6. \(\log _{2} 5-\log _{2}(x-1)\)
    7. \(\log _{7} 3+\log _{7} x-\log _{7} y\)
    8. \(\log _{5} 2-\log _{5} x-\log _{5} y\)
    9. \(4 \log _{2} x+6 \log _{2} y\)
    10. \(6 \log _{3} x+9 \log _{3} y\)
    11. \(\log _{3}\left(x^{2}-1\right)-2 \log _{3}(x-1)\)
    12. \(\log \left(x^{2}+2 x+1\right)-2 \log (x+1)\)
    13. \(4 \log x-2 \log y-3 \log z\)
    14. \(3 \ln x+4 \ln y-2 \ln z\)
    15. \(\frac{1}{3} \log x-3 \log (x+1)\)
    16. \(2 \log (2 x+3)+\frac{1}{2} \log (x+1)\)
    Jibu

    2. \(2\)

    4. \(2\)

    6. \(\log _{2} \frac{5}{x-1}\)

    8. \(\log _{5} \frac{2}{x y}\)

    10. \(\log _{3} x^{6} y^{9}\)

    12. \(0\)

    14. \(\ln \frac{x^{3} y^{4}}{z^{2}}\)

    16. \(\log (2 x+3)^{2} \cdot \sqrt{x+1}\)

    Zoezi\(\PageIndex{27}\) Use the Change-of-Base Formula

    Katika mazoezi yafuatayo, tumia Mfumo wa Mabadiliko ya Msingi, ukizunguka kwenye maeneo matatu ya decimal, ili takriban kila logarithm.

    1. \(\log _{3} 42\)
    2. \(\log _{5} 46\)
    3. \(\log _{12} 87\)
    4. \(\log _{15} 93\)
    5. \(\log _{\sqrt{2}} 17\)
    6. \(\log _{\sqrt{3}} 21\)
    Jibu

    2. \(2.379\)

    4. \(1.674\)

    6. \(5.542\)

    Zoezi\(\PageIndex{28}\) Writing Exercises
    1. Andika Mali ya Bidhaa kwa maneno yako mwenyewe. Je, inatumika kwa kila moja ya yafuatayo? \(\log _{a} 5 x, \log _{a}(5+x)\). Kwa nini au kwa nini?
    2. Andika Mali ya Nguvu kwa maneno yako mwenyewe. Je, inatumika kwa kila moja ya yafuatayo? \(\log _{a} x^{p},\left(\log _{a} x\right)^{r}\). Kwa nini au kwa nini?
    3. Tumia mfano ili kuonyesha kwamba\(\log (a+b) \neq \log a+\log b ?\)
    4. Eleza jinsi ya kupata thamani ya\(\log _{7} 15\) kutumia 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 tatu 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 kutumia mali ya logarithms na kutumia mabadiliko ya formula ya msingi. Wengine wa seli ni tupu.
    Kielelezo 10.4.5

    b. kwa kiwango cha\(1−10\), jinsi gani unaweza kiwango mastery yako ya sehemu hii katika mwanga wa majibu yako katika orodha? Unawezaje kuboresha hii?