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Sura ya 10 Mazoezi Mapitio

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    Sura ya Mapitio ya mazoezi

    Kutafuta Kazi za Composite na Inverse

    Zoezi\(\PageIndex{1}\) Find and Evaluate Composite Functions

    Katika mazoezi yafuatayo, kwa kila jozi ya kazi, tafuta

    1. \((f \circ g)(x)\)
    2. \((g \circ f)(x)\)
    3. \((f \cdot g)(x)\)

    1. \(f(x)=7 x-2\)na\(g(x)=5 x+1\)

    2. \(f(x)=4 x\)na\(g(x)=x^{2}+3 x\)

    Jibu

    2.

    1. \(4 x^{2}+12 x\)
    2. \(16 x^{2}+12 x\)
    3. \(4 x^{3}+12 x^{2}\)
    Zoezi\(\PageIndex{2}\) Find and Evaluate Composite Functions

    Katika mazoezi yafuatayo, tathmini muundo.

    1. Kwa kazi\(f(x)=3 x^{2}+2\) na\(g(x)=4 x-3\), tafuta
      1. \((f \circ g)(-3)\)
      2. \((g \circ f)(-2)\)
      3. \((f \circ f)(-1)\)
    2. Kwa kazi\(f(x)=2 x^{3}+5\) na\(g(x)=3 x^{2}-7\), tafuta
      1. \((f \circ g)(-1)\)
      2. \((g \circ f)(-2)\)
      3. \((g \circ g)(1)\)
    Jibu

    2.

    1. \(-123\)
    2. \(356\)
    3. \(41\)
    Zoezi\(\PageIndex{3}\) Determine Whether a Function is One-to-One

    Katika mazoezi yafuatayo, kwa kila seti ya jozi zilizoamriwa, onyesha ikiwa inawakilisha kazi na ikiwa ni hivyo, ni kazi moja kwa moja.

    1. \(\begin{array}{l}{\{(-3,-5),(-2,-4),(-1,-3),(0,-2)} , {(-1,-1),(-2,0),(-3,1) \}}\end{array}\)
    2. \(\begin{array}{l}{\{(-3,0),(-2,-2),(-1,0),(0,1)} , {(1,2),(2,1),(3,-1) \}}\end{array}\)
    3. \(\begin{array}{l}{\{(-3,3),(-2,1),(-1,-1),(0,-3)} , {(1,-5),(2,-4),(3,-2) \}}\end{array}\)
    Jibu

    2. Kazi; si moja kwa moja

    Zoezi\(\PageIndex{4}\) Determine Whether a Function is One-to-One

    Katika mazoezi yafuatayo, onyesha kama kila grafu ni grafu ya kazi na ikiwa ni hivyo, ni moja kwa moja.


      1. Takwimu hii inaonyesha mstari kutoka (hasi 6, hasi 2) hadi (hasi 1, 3) na kisha chini kutoka hapo hadi (6, hasi 4).
        Kielelezo 10.E.1

      2. Takwimu hii inaonyesha mstari kutoka (6, 5) chini ya (0, hasi 1) na kisha chini kutoka huko hadi (5, hasi 6).
        Kielelezo 10.E.2

      1. Takwimu hii inaonyesha mstari wa mviringo kutoka (hasi 6, hasi 2) hadi asili na kisha kuendelea kutoka huko hadi (6, 2).
        Kielelezo 10.E.3

      2. Takwimu hii inaonyesha mduara wa radius 2 na kituo cha asili.
        Kielelezo 10.E.4
    Jibu

    1.

    1. Kazi; si moja kwa moja
    2. Si kazi
    Zoezi\(\PageIndex{5}\) Find the Inverse of a Function

    Katika zoezi zifuatazo, tafuta inverse ya kazi. Tambua kikoa na upeo wa kazi ya inverse.

    1. \(\{(-3,10),(-2,5),(-1,2),(0,1)\}\)
    Jibu

    1. Inverse kazi:\(\{(10,-3),(5,-2),(2,-1),(1,0)\}\). Domain:\(\{1,2,5,10\}\). Mipangilio:\(\{-3,-2,-1,0\}\).

    Zoezi\(\PageIndex{6}\) Find the Inverse of a Function

    Katika zoezi zifuatazo, graph inverse ya kazi moja kwa moja iliyoonyeshwa.

    Takwimu hii inaonyesha sehemu ya mstari kutoka (hasi 4, hasi 2) hadi (hasi 2, 1) kisha hadi (2, 2) na kisha hadi (3, 4).
    Kielelezo 10.E.5
    Jibu

    Tatua peke yako

    Zoezi\(\PageIndex{7}\) Find the Inverse of a Function

    Katika mazoezi yafuatayo, hakikisha kwamba kazi ni kazi za inverse.

    1. \(\begin{array}{l}{f(x)=3 x+7 \text { and }} {g(x)=\frac{x-7}{3}}\end{array}\)
    2. \(\begin{array}{l}{f(x)=2 x+9 \text { and }} {g(x)=\frac{x+9}{2}}\end{array}\)
    Jibu

    1. \(g(f(x))=x,\)na\(f(g(x))=x,\) hivyo wao ni inverses.

    Zoezi\(\PageIndex{8}\) Find the Inverse of a Function
    1. \(f(x)=6 x-11\)
    2. \(f(x)=x^{3}+13\)
    3. \(f(x)=\frac{1}{x+5}\)
    4. \(f(x)=\sqrt[5]{x-1}\)
    Jibu

    1. \(f^{-1}(x)=\frac{x+11}{6}\)

    3. \(f^{-1}(x)=\frac{1}{x}-5\)

    Tathmini na Grafu Kazi za Kielelezo

    Zoezi\(\PageIndex{9}\) Graph Exponential Functions

    Katika mazoezi yafuatayo, graph kila moja ya kazi zifuatazo.

    1. \(f(x)=4^{x}\)
    2. \(f(x)=\left(\frac{1}{5}\right)^{x}\)
    3. \(g(x)=(0.75)^{x}\)
    4. \(g(x)=3^{x+2}\)
    5. \(f(x)=(2.3)^{x}-3\)
    6. \(f(x)=e^{x}+5\)
    7. \(f(x)=-e^{x}\)
    Jibu

    1.

    Takwimu hii inaonyesha mstari wa kielelezo unaopitia pointi (hasi 1, 1 juu ya 4), (0, 1), na (1, 4).
    Kielelezo 10.E.6

    3.

    Takwimu hii inaonyesha mstari wa kielelezo unaopitia pointi (hasi 1, 4 juu ya 3), (0, 1), na (1, 3 juu ya 4).
    Kielelezo 10.E.7

    5.

    Takwimu hii inaonyesha mstari wa kielelezo unaopitia pointi (hasi 1, hasi 59 juu ya 23), (0, hasi 2), na (1, hasi 7 juu ya 10).
    Kielelezo 10.E.8

    7.

    Takwimu hii inaonyesha mstari wa kielelezo unaopitia pointi (hasi 1, hasi 1 juu ya e), (0, hasi 1), na (1, hasi e).
    Kielelezo 10.E.9
    Zoezi\(\PageIndex{10}\) Solve Exponential Equations

    Katika mazoezi yafuatayo, tatua kila equation.

    1. \(3^{5 x-6}=81\)
    2. \(2^{x^{2}}=16\)
    3. \(9^{x}=27\)
    4. \(5^{x^{2}+2 x}=\frac{1}{5}\)
    5. \(e^{4 x} \cdot e^{7}=e^{19}\)
    6. \(\frac{e^{x^{2}}}{e^{15}}=e^{2 x}\)
    Jibu

    2. \(x=-2, x=2\)

    4. \(x=-1\)

    6. \(x=-3, x=5\)

    Zoezi\(\PageIndex{11}\) Use Exponential Models in Applications

    Katika mazoezi yafuatayo, tatua.

    1. Felix imewekeza $\(12,000\) katika akaunti ya akiba. Ikiwa kiwango cha riba ni\(4\)% kiasi gani kitakuwa katika akaunti kwa\(12\) miaka kwa kila njia ya kuimarisha?
      1. kiwanja robo mwaka
      2. kiwanja kila mwezi
      3. kiwanja kuendelea
    2. Sayed amana $\(20,000\) katika akaunti ya uwekezaji. Nini itakuwa thamani ya uwekezaji wake katika\(30\) miaka kama uwekezaji ni kupata\(7\)% kwa mwaka na ni imezungukwa kuendelea?
    3. Mtafiti katika Kituo cha Kudhibiti na Kuzuia Magonjwa anasoma ukuaji wa bakteria. Anaanza majaribio yake na\(150\) ya bakteria ambayo inakua kwa kiwango cha\(15\)% kwa saa. Ataangalia bakteria kila\(24\) masaa. Ni bakteria ngapi atakayopata kwa\(24\) masaa?
    4. Katika miaka mitano iliyopita idadi ya watu wa Marekani imeongezeka kwa kiwango cha\(0.7\)% kwa mwaka hadi karibu\(318,900,000\). Kama kiwango hiki kinaendelea, nini itakuwa idadi ya watu katika miaka\(5\) zaidi?
    Jibu

    2. \(\$ 163,323.40\)

    4. \(330,259,000\)

    Tathmini na Grafu Kazi za Logarithmic

    Zoezi\(\PageIndex{12}\) Convert Between Exponential and Logarithmic Form

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

    1. \(5^{4}=625\)
    2. \(10^{-3}=\frac{1}{1,000}\)
    3. \(63^{\frac{1}{5}}=\sqrt[5]{63}\)
    4. \(e^{y}=16\)
    Jibu

    2. \(\log \frac{1}{1,000}=-3\)

    4. \(\ln 16=y\)

    Zoezi\(\PageIndex{13}\) Convert Between Exponential and Logarithmic Form

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

    1. \(7=\log _{2} 128\)
    2. \(5=\log 100,000\)
    3. \(4=\ln x\)
    Jibu

    2. \(100000=10^{5}\)

    Zoezi\(\PageIndex{14}\) Evaluate Logarithmic Functions

    Katika mazoezi yafuatayo, tatua\(x\).

    1. \(\log _{x} 125=3\)
    2. \(\log _{7} x=-2\)
    3. \(\log _{\frac{1}{2}} \frac{1}{16}=x\)
    Jibu

    1. \(x=5\)

    3. \(x=4\)

    Zoezi\(\PageIndex{15}\) Evaluate Logarithmic Functions

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

    1. \(\log _{2} 32\)
    2. \(\log _{8} 1\)
    3. \(\log _{3} \frac{1}{9}\)
    Jibu

    2. \(0\)

    Zoezi\(\PageIndex{16}\) Graph Logarithmic Functions

    Katika mazoezi yafuatayo, grafu kila kazi ya logarithmic.

    1. \(y=\log _{5} x\)
    2. \(y=\log _{\frac{1}{4}} x\)
    3. \(y=\log _{0.8} x\)
    Jibu

    1.

    Takwimu hii inaonyesha mstari wa logarithmic unaopitia pointi (1 juu ya 5, hasi 1), (1, 0), na (5, 1).
    Kielelezo 10.E.10

    3.

    Takwimu hii inaonyesha mstari wa logarithmic unaopitia pointi (4 juu ya 5, 1), (1, 0), na (5 juu ya 4, hasi 1).
    Kielelezo 10.E.11
    Zoezi\(\PageIndex{17}\) Solve Logarithmic Equations

    Katika mazoezi yafuatayo, tatua kila equation ya logarithmic.

    1. \(\log _{a} 36=5\)
    2. \(\ln x=-3\)
    3. \(\log _{2}(5 x-7)=3\)
    4. \(\ln e^{3 x}=24\)
    5. \(\log \left(x^{2}-21\right)=2\)
    Jibu

    2. \(x=e^{-3}\)

    4. \(x=8\)

    Zoezi\(\PageIndex{18}\) Use Logarithmic Models in Applications

    Je, ni kiwango cha decibel cha filimbi cha treni na\(10^{−3}\) watts kali kwa inchi ya mraba?

    Jibu

    \(90\)dB

    Tumia Mali ya Logarithms

    Zoezi\(\PageIndex{19}\) Use the Properties of Logarithms

    Katika mazoezi yafuatayo, tumia mali ya logarithms kutathmini.

      1. \(\log _{7} 1\)
      2. \(\log _{12} 12\)
      1. \(5^{\log _{5} 13}\)
      2. \(\log _{3} 3^{-9}\)
      1. \(10^{\log \sqrt{5}}\)
      2. \(\log 10^{-3}\)
      1. \(e^{\ln 8}\)
      2. \(\ln e^{5}\)
    Jibu

    2.

    1. \(13\)
    2. \(-9\)

    4.

    1. \(8\)
    2. \(5\)
    Zoezi\(\PageIndex{20}\) 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. \(\log _{4}(64 x y)\)
    2. \(\log 10,000 m\)
    Jibu

    2. \(4+\log m\)

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

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

    1. \(\log _{7} \frac{49}{y}\)
    2. \(\ln \frac{e^{5}}{2}\)
    Jibu

    2. \(5-\ln 2\)

    Zoezi\(\PageIndex{22}\) Use the Properties of Logarithms

    Katika mazoezi yafuatayo, tumia Mali ya Nguvu ya Logarithms kupanua kila logarithm. Kurahisisha, ikiwa inawezekana.

    1. \(\log x^{-9}\)
    2. \(\log _{4} \sqrt[7]{z}\)
    Jibu

    2. \(\frac{1}{7} \log _{4} z\)

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

    Katika mazoezi yafuatayo, tumia mali ya logarithms kuandika kila logarithm kama jumla ya logarithms. Kurahisisha kama inawezekana.

    1. \(\log _{3}\left(\sqrt{4} x^{7} y^{8}\right)\)
    2. \(\log _{5} \frac{8 a^{2} b^{6} c}{d^{3}}\)
    3. \(\ln \frac{\sqrt{3 x^{2}-y^{2}}}{z^{4}}\)
    4. \(\log _{6} \sqrt[3]{\frac{7 x^{2}}{6 y^{3} z^{5}}}\)
    Jibu

    2. \(\begin{array}{l}{\log _{5} 8+2 \log _{5} a+6 \log _{5} b} {+\log _{5} c-3 \log _{5} d}\end{array}\)

    4. \(\begin{array}{l}{\frac{1}{3}\left(\log _{6} 7+2 \log _{6} x-1-3 \log _{6} y\right.} {-5 \log _{6} z )}\end{array}\)

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

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

    1. \(\log _{2} 56-\log _{2} 7\)
    2. \(3 \log _{3} x+7 \log _{3} y\)
    3. \(\log _{5}\left(x^{2}-16\right)-2 \log _{5}(x+4)\)
    4. \(\frac{1}{4} \log y-2 \log (y-3)\)
    Jibu

    2. \(\log _{3} x^{3} y^{7}\)

    4. \(\log \frac{\sqrt[4]{y}}{(y-3)^{2}}\)

    Zoezi\(\PageIndex{25}\) Use the Change-of-Base Formula

    Katika mazoezi yafuatayo, kuzunguka maeneo matatu ya decimal, takriban kila logarithm.

    1. \(\log _{5} 97\)
    2. \(\log _{\sqrt{3}} 16\)
    Jibu

    2. \(5.047\)

    Tatua Ulinganisho wa Kielelezo na Logarithmic

    Zoezi\(\PageIndex{26}\) Solve Logarithmic Equations Using the Properties of Logarithms

    Katika mazoezi yafuatayo, tatua\(x\).

    1. \(3 \log _{5} x=\log _{5} 216\)
    2. \(\log _{2} x+\log _{2}(x-2)=3\)
    3. \(\log (x-1)-\log (3 x+5)=-\log x\)
    4. \(\log _{4}(x-2)+\log _{4}(x+5)=\log _{4} 8\)
    5. \(\ln (3 x-2)=\ln (x+4)+\ln 2\)
    Jibu

    2. \(x=4\)

    4. \(x=3\)

    Zoezi\(\PageIndex{27}\) 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. \(2^{x}=101\)
    2. \(e^{x}=23\)
    3. \(\left(\frac{1}{3}\right)^{x}=7\)
    4. \(7 e^{x+3}=28\)
    5. \(e^{x-4}+8=23\)
    Jibu

    1. \(x=\frac{\log 101}{\log 2} \approx 6.658\)

    3. \(x=\frac{\log 7}{\log \frac{1}{3}} \approx-1.771\)

    5. \(x=\ln 15+4 \approx 6.708\)

    Zoezi\(\PageIndex{28}\) Use Exponential Models in Applications
    1. Jerome inawekeza $\(18,000\) akiwa na umri\(17\). Anatumaini uwekezaji utakuwa na thamani ya $\(30,000\) wakati anarudi\(26\). Ikiwa maslahi huchanganya kuendelea, takriban kiwango gani cha ukuaji atahitaji kufikia lengo lake? Je, hiyo ni matarajio ya kuridhisha?
    2. Elise anawekeza $\(4500\) katika akaunti ambayo huchanganya maslahi ya kila mwezi na\(6\) hupata% .Itachukua muda gani kwa pesa yake mara mbili?
    3. Watafiti walirekodi kuwa idadi fulani ya bakteria ilikua kutoka\(100\)\(300\) kwa\(8\) saa. Kwa kiwango hiki cha ukuaji, ni bakteria ngapi zitakavyokuwa na\(24\) masaa?
    4. Idadi ya watu wanaweza mara mbili kwa\(8\) miezi\(\left(A=2 A_{0}\right)\). Itachukua muda gani kwa idadi ya panya kwa mara tatu?
    5. Maisha ya nusu ya iodini ya mionzi ni\(60\) siku. Ni kiasi gani cha sampuli ya\(50\) mg kitaachwa kwa\(40\) siku?
    Jibu

    2. \(11.6\)miaka

    4. \(12.7\)miezi

    Mazoezi mtihani

    Zoezi\(\PageIndex{29}\)
    1. Kwa kazi,\(f(x)=6x+1\) na\(g(x)=8x−3\), tafuta
      1. \((f \circ g)(x)\)
      2. \((g \circ f)(x)\)
      3. \((f \cdot g)(x)\)
    2. Kuamua kama seti zifuatazo za jozi kuamuru inawakilisha kazi na kama ni hivyo, ni kazi moja kwa moja. \(\{(-2,2),(-1,-3),(0,1),(1,-2),(2,-3)\}\)
    3. Kuamua kama kila grafu ni grafu ya kazi na kama ni hivyo, ni moja kwa moja.

      1. Takwimu hii inaonyesha ufunguzi wa parabola kwa haki na vertex (hasi 3, 0).
        Kielelezo 10.E.12

      2. Takwimu hii inaonyesha mstari wa kielelezo unaopitia pointi (hasi 1, 1 juu ya 2), (0, 1), na (1, 2).
        Kielelezo 10.E.13
    4. Grafu, kwenye mfumo huo wa kuratibu, inverse ya kazi moja kwa moja iliyoonyeshwa.
    Takwimu hii inaonyesha sehemu ya mstari inayopita kutoka hatua (hasi 3, 3) hadi (hasi 1, 2) hadi (0, hasi 2) hadi (2, hasi 4).
    Kielelezo 10.E.14

    5. Pata inverse ya kazi\(f(x)=x^{5}−9\).

    6. Graph kazi\(g(x)=2^{x-3}\).

    7. Kutatua equation\(2^{2 x-4}=64\).

    8. Kutatua equation\(\frac{e^{x^{2}}}{e^{4}}=e^{3 x}\).

    9. Megan imewekeza $\(21,000\) katika akaunti ya akiba. Ikiwa kiwango cha riba ni\(5\)%, ni kiasi gani kitakuwa katika akaunti kwa\(8\) miaka kwa kila njia ya kuimarisha?

    1. kiwanja robo mwaka
    2. kiwanja kila mwezi
    3. kiwanja kuendelea

    10. Badilisha equation kutoka kielelezo kwa fomu ya logarithmic:\(10^{-2}=\frac{1}{100}\).

    11. Badilisha equation kutoka equation logarithmic kwa fomu kielelezo:\(3=\log _{7} 343\).

    12. Tatua kwa\(x\):\(\log _{5} x=-3\)

    13. Tathmini logi\(_{11} 1\).

    14. Tathmini\(\log _{4} \frac{1}{64}\).

    15. Graph kazi\(y=\log _{3} x\).

    16. Tatua kwa\(x\):\(\log \left(x^{2}-39\right)=1\)

    17. Ngazi ya decibel ya shabiki mdogo na\(10^{−8}\) watts kali kwa inchi ya mraba ni nini?

    18. Tathmini kila mmoja.

    1. \(6^{\log _{6} 17}\)
    2. \(\log _{9} 9^{-3}\)
    Jibu

    1.

    1. \(48 x-17\)
    2. \(48 x+5\)
    3. \(48 x^{2}-10 x-3\)

    3.

    1. Si kazi
    2. Kazi moja kwa moja

    5. \(f^{-1}(x)=\sqrt[5]{x+9}\)

    7. \(x=5\)

    9.

    1. $\(31,250.74\)
    2. $\(31,302.29\)
    3. $\(31,328.32\)

    11. \(343=7^{3}\)

    13. \(0\)

    15.

    Takwimu hii inaonyesha mstari wa logarithmic unaopita (1 juu ya 3, 1), (1, 0), na (3, 1).
    Kielelezo 10.E.15

    17. \(40\)dB

    Zoezi\(\PageIndex{30}\)

    Katika mazoezi yafuatayo, tumia mali ya logarithms kuandika kila kujieleza kama jumla ya logarithms, kurahisisha iwezekanavyo.

    1. \(\log _{5} 25 a b\)
    2. \(\ln \frac{e^{12}}{8}\)
    3. \(\log _{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}\)
    Jibu

    1. \(2+\log _{5} a+\log _{5} b\)

    3. \(\begin{array}{l}{\frac{1}{4}\left(\log _{2} 5+3 \log _{2} x-4-2 \log _{2} y\right.} {-7 \log _{2} z )}\end{array}\)

    Zoezi\(\PageIndex{31}\)

    Katika mazoezi yafuatayo, tumia Mali ya Logarithms ili kuimarisha logarithm, kurahisisha iwezekanavyo.

    1. \(5 \log _{4} x+3 \log _{4} y\)
    2. \(\frac{1}{6} \log x-3 \log (x+5)\)
    3. Kuzunguka kwa maeneo matatu ya decimal, takriban\(\log _{4} 73\).
    4. Tatua kwa\(x\):\(\log _{7}(x+2)+\log _{7}(x-3)=\log _{7} 24\)
    Jibu

    2. \(\log \frac{\sqrt[6]{x}}{(x+5)^{3}}\)

    4. \(x=6\)

    Zoezi\(\PageIndex{32}\)

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

    1. \(\left(\frac{1}{5}\right)^{x}=9\)
    2. \(5 e^{x-4}=40\)
    3. Jacob uwekezaji $\(14,000\) katika akaunti ambayo misombo riba robo mwaka na chuma\(4\)%. Itachukua muda gani kwa pesa zake mara mbili?
    4. Watafiti walirekodi kuwa idadi fulani ya bakteria ilikua kutoka\(500\)\(700\) kwa\(5\) saa. Kwa kiwango hiki cha ukuaji, ni bakteria ngapi zitakavyokuwa na\(20\) masaa?
    5. Idadi fulani ya beetle inaweza mara mbili kwa\(3\) miezi\(\left(A=2 A_{0}\right)\). Itachukua muda gani kwa idadi ya mende kuwa mara tatu?
    Jibu

    2. \(x=\ln 8+4 \approx 6.079\)

    4. \(1,921\)bakteria