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The Logarithmic Function Quiz

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This quiz tests the work covered in Lecture 3 and corresponds to Section 1.4 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).

There are two sections of economics quiz site that would be useful, Logarithmic and Square Root Graphs and Logarithmic Functions.

The Mathematics Learning Centre has a booklet on Introduction to Exponents and Logarithms and tutors who can help you with the concepts.

There is a further useful site at http://www.ping.be/ ping1339/exp.htm which considers both logarithmic and exponential functions, as does http://www.themathpage.com/aPreCalc/logarithmic-exponential-functions.htm. There is a useful animation at http://www.analyzemath.com/logfunction/logfunction.html which shows how logarithmic functions change with different parameters.

Question 1

Which of the following statements are correct? There may be more than one correct answer.

a)
2.35 e = 10.49 ⇒ ln10.49 = 2.35.
b)
2.35 e = 10.49 ⇒ ln 2.35 = 10.49.
c)
1.23 log1017 = 1.23 ⇒ 10 = 17.
d)
10 log10 17 = 1.23 ⇒ 17 = 1.23.

 

There is at least one mistake.
For example, choice (a) should be true.
There is at least one mistake.
For example, choice (b) should be false.
Note that c e = x ⇒ lnx = c.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be false.
Note that c log10x = c ⇒ 10 = x.
Your answers are correct
  1. True.
  2. False. Note that c e = x ⇒ lnx = c.
  3. True.
  4. False. Note that c log10x = c ⇒ 10 = x.

Question 2

Consider the equation  3e2x = 5e4x.
Which of the following solve this equation?

a)
e2 = 0.6.
  b)
x = - 0.0766.
c)
x = - 0.3065.
  d)
x = 1.0217.

 

Not correct. Choice (a) is false.
Try again, you have not used the index laws correctly.
Your answer is correct.
Rearrange the equation to become e4x 3 e2x = 5 = 0.6
then e4x-2x = e2x = 0.6, so we take the natural log of both sides
2x ln0.6 ln(e ) = 2x = ln0.6 ⇒ x = 2 = - 0.0766.
Not correct. Choice (c) is false.
Try again, you may have tried to solve e0.5x = 3. 5
Not correct. Choice (d) is false.
Try again, you may have tried to solve 5 e2x = 3.

Question 3

Which of the following are written correctly as simpler logarithmic quantities?
There may be more than one correct answer.

a)
( x+ 1 ) ln x+-2- = ln(x+ 1)+ ln(x = 2).
b)
√ ------ 1 ln 3x+ 1 = 2 ln(3x+ 1).
c)
√----- ( x2-x-+-1) 2 √ ----- √3------ ln √33x-+-4- = ln x + ln x+ 1- ln 3x+ 4.
d)
( x2√x-+-1) 1 1 ln √33x-+-4- = 2lnx + 2 ln(x+ 1)+ 3 ln(3x+ 4).

 

There is at least one mistake.
For example, choice (a) should be false.
Recall ( a) ln b-= lna - ln b.
There is at least one mistake.
For example, choice (b) should be true.
√------ 1 b 3x + 1 = (3x +1)2and lna = bln a.
There is at least one mistake.
For example, choice (c) should be true.
(a-) b ln b = lna - lnb and ln a = blna.
There is at least one mistake.
For example, choice (d) should be false.
Recall (a) ln b- = lna- lnb.
Your answers are correct
  1. False. Recall ( a) ln b-= lna - ln b.
  2. True. √------ 1 b 3x + 1 = (3x +1)2and lna = bln a.
  3. True. (a-) b ln b = lna - lnb and ln a = blna.
  4. False. Recall (a) ln b- = lna- lnb.

Question 4

The population of a city increases at a rate which is proportional to the current population and was 2 million in 1980 and 2.5 million in 1990.
Which of the following gives the population, P (in millions), t years after 1980?

a)
-0.022t P (t) = 2.5e .
  b)
-0.022t P(t) = 2e .
c)
0.022t P (t) = 2.5e .
  d)
0.022t P(t) = 2e .

 

Not correct. Choice (a) is false.
Try again, you seem to have confused the initial population and the population after 10 years.
Not correct. Choice (b) is false.
Try again, you seem to have confused the initial population and the population after 10 years in your working.
Not correct. Choice (c) is false.
Try again, you seem to have confused the initial population and the population after 10 years.
Your answer is correct.
Since P increases at a rate which is proportional to the current population
kt P = P0e where P0 = 2 million, the population in 1980.
We have that 10k 2.5 = 2e since the population is 2.5 million after 10 years.
Solving this for k we have 10k 10k e = 1.25 ⇒ ln(e ) = 10k = ln1.25
So k = ln-1.25-= 0.022 10 and P (t) = 2e0.022t.
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