2.14 · Logarithmic Function Context and Data Modeling

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1. 1a.

   Mr. Bean is learning Chinese. Model # of days studying as a function of # of words learned.
   (a) What is a logarithmic regression equation that could model this situation?

   # of Words Learned	10	32	79	197	400
   # of Days Studying	1	6	10	14	17

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2. 1b.

   (1) How many days would it take for Mr. Bean to learn 1,000 words?
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3. 1c.

   (1) How many words would he learn if he studied for 2 months (60 days)?
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4. 2a.

   Mr. Sullivan's diet: pounds lost in a week as a function of week number.
   (a) What's a logarithmic regression equation that could model this situation?

   Week #	1	2	6	9
   Pounds lost that week	9.5	8.5	6.5	5.5

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5. 2b.

   (2) During which week will Mr. Sullivan lose 2 pounds?
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6. 2c.

   (2) How many pounds will Mr. Sullivan lose during the 12th week?
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7. 3a.

   Math Geek Tech stock price (dollars) as a function of days on the market.
   (a) What's a logarithmic regression equation that could model this situation?

   Days on Market	1	3	4	7
   Stock Prices ($)	112	150.5	160.5	180.1

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8. 3b.

   (3) How many days will it take for the stock price to be $250?
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9. 3c.

   (3) What would the stock price be after the first two weeks on the market (14 days)?
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10. 4a.

    Population model $N(t)=N_0e^{rt}$ ($N_0$ = initial pop., $r$ = growth rate, $t$ = years). Round to 3 decimals.
    (a) Population of Cleveland, OH in 10 years if there are currently 275,000 people, growth rate 2.5%?
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11. 4b.

    (4) What growth rate would Cleveland need to reach 400,000 people in 20 years?
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12. 5a.

    Newton's Law of Cooling: $t=-10\ln\left(\frac{T-R}{98.6-R}\right)$, $T$ = body temp (°F), $R$ = room temp, $t$ = hours since death.
    (a) How many hours elapsed if the room was 75° and the body's temp was 85°?
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13. 5b.

    (5) A body was found 5 hours after death in a 65° room. What was the body's temperature when found?
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14. 6.

    A logarithmic regression is used to model the data. What is the value of $f(4.5)$?
    (A) 35.225 (B) 34.987 (C) 54.639 (D) 54.012

    $x$	$f(x)$
    1	44.9
    3	37.9
    7	32.4
    9	30.7

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15. 7A(i).

    Let $f$ be increasing, defined for $x>0$ (table). $g(x)=\frac{x^2-3x+5}{x-3}$.
    (A)(i) $h(x)=g(f(x))$. Find $h(4)$ as a decimal, or indicate it is not defined.

    $x$	4	8	16	32	64
    $f(x)$	1	5	9	13	17

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16. 7A(ii).

    (7) (A)(ii) Find $f^{-1}(64)$, or indicate it is not defined.
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17. 7B(i).

    (7) (B)(i) Find all zeros of $g(x)$, as decimals, or indicate there are none.
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18. 7B(ii).

    (7) (B)(ii) Determine the end behavior of $g$ as $x$ decreases without bound (limit notation).
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19. 7C(i).

    (7) (C)(i) Use the table to determine if $f$ is best modeled by a linear, quadratic, exponential, or logarithmic function.
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20. 7C(ii).

    (7) (C)(ii) Give a reason based on the relationship between the change in outputs and the change in inputs.
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