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Quiz 5: Properties of Logs and Exponentials

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Which of the following are defined for all real values of x  ?
(1) log10ex (2) (x2 +1)x (3) (x2 - 1)x
a) (1) and (2) and (3)   b) (2) and (3) only.
c) (1) and (2) only.   d) (1) and (3) only.
e) None of them

 

Not correct. Choice (a) is false.
Remember that b blna a = e and so b a makes sense only when a > 0 .
Not correct. Choice (b) is false.
Remember that b blna a = e and so b a makes sense only when a > 0 .
Your answer is correct.
Not correct. Choice (d) is false.
Remember that b blna a = e and so b a makes sense only when a > 0 .
Not correct. Choice (e) is false.
 

Question 2

 
 
Which option is a simplified version of the expression ln(5e2x)+ eln(5x)  ?
a) ln10+ lnx + 5x , for all x > 0   b) ln5 + 10x , for all x
c) ln 2+ ln x+ ln5 , for all x > 0   d) 7x + ln 5 , for all x
e) None of the above.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Your answer is correct.
The expression only makes sense when x > 0 and so the answer is 7x+ ln5 , for all x > 0
 

Question 3

 
 
If y = ln(√e2x +-4) , what is x in terms of y  ?
a) x = 1 ln(e2y - 4) 2   b) x = (ln(e2y - 4))2
c) x = 1(2y - ln 4) 2   d) x = 1ln(ey - 4) 4
e) x = 1ln(e2y - 2) 2

 

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 4

 
 
If y = 3xex , select the option which equals dy. dx
a) 3xex(x + 1)ex ln 3   b) x x (xe + e )ln3
c) x xex- 1 xe 3   d) x x+ex-2 xe 3
e) None of the above

 

Your answer is correct.
Not correct. Choice (b) is false.
Try writing y as an exponential, or alternatively, take logs of both sides and use logarithmic differentiation.
Not correct. Choice (c) is false.
Try writing y as an exponential, or alternatively, take logs of both sides and use logarithmic differentiation.
Not correct. Choice (d) is false.
Try writing y as an exponential, or alternatively, take logs of both sides and use logarithmic differentiation.
Not correct. Choice (e) is false.
 

Question 5

 
 
Use logarithmic differentiation to find dy dx- when √-3----4 (ln(sin x))2 y = x +x -1-+sinx--
a) (3x2 + 4x3 cosx ) y 2(x3 +-x4) +2 cot x- 1-+sinx   b) ( ) 3x2 +-4x3 cotx-- --cosx--- y (x3 + x4) + lnsinx - 1+ sinx
c) ( 2 3 ) y 3x-+-4x--+ 2-cotx-- ---1---- 2(x3 + x4) lnsin x 1+ sinx   d) ( 2 3 ) y 3x3+-4x4-+ 2-cotx-- -cosx--- 2(x + x ) lnsin x 1+ sinx
e) None of the above

 

Not correct. Choice (a) is false.
Taking natural logs of both sides of the equation for y gives
lny = 1 ln(x3 + x4)+ 2ln(ln sinx))- ln(1+ sin x). 2
Now differentiate both sides with respect to x .
Not correct. Choice (b) is false.
Taking natural logs of both sides of the equation for y gives
1 lny = - ln(x3 + x4)+ 2ln(ln sinx))- ln(1+ sin x). 2
Now differentiate both sides with respect to x .
Not correct. Choice (c) is false.
Taking natural logs of both sides of the equation for y gives
1 3 4 lny = 2 ln(x + x )+ 2ln(ln sinx))- ln(1+ sin x).
Now differentiate both sides with respect to x .
Your answer is correct.
Not correct. Choice (e) is false.
 

Question 6

 
 
Which values of t satisfy simultaneously the pair of inequalities ∣ ln∣t∣ ∣< 1 and √t2-> 1  ?
a) All t such that 1 < t < e .   b) All t in the intervals (- e,- 1) or (1,e) .
c) All t such that 1 e < t < e .   d) All t ≥ 1 .
e) All t such that 1 - e < t < -e or 1 < t < e e

 

Not correct. Choice (a) is false.
Remember that the solution set of √t2-> 1 will contain negative as well as positive numbers!
Your answer is correct.
Not correct. Choice (c) is false.
Remember that if ∣ ln∣t∣ ∣< 1 , then - 1 < ln∣t∣ < 1.
Not correct. Choice (d) is false.
Remember that if ∣ ln∣t∣ ∣< 1 , then - 1 < ln ∣t∣ < 1 and that the solution set of √ -- t2 > 1 will contain negative as well as positive numbers!
Not correct. Choice (e) is false.
Remember that if ∣ ln∣t∣ ∣< 1 , then - 1 < ln∣t∣ < 1 and that the solution set of √-- t2 > 1 will contain negative as well as positive numbers!
 

Question 7

 
 
If y = 3-x - 22x and z = 32x + 2-x , what is yz  ?
a) 3x + 2x - 62x +6- x   b) 3x - 2x + 6-x
c) 6- 2x - 64x   d) 3x - 2x + 6x
e) None of the above

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Your answer is correct.
In fact, yz = 3x - 2x + 6-x(1- 63x) .
 

Question 8

 
 
Solve the equation √ -- ln( ex - 1) = 0 for x and enter your answer correct to two decimal places.

 

Your answer is correct
Taking exponentials gives √ -- ex - 1 = 1 , after which we obtain √ -- ex = 2, and then x = ln 4 .
Not correct. You may try again.
Have you taken exponentials of each side of the equation? This gives √-- ex - 1 = 1 , from which you can obtain the unique solution for x after further manipulations.
 

Question 9

 
 
Two of the following have identical derivatives. Tick the pair that do.
a) 6x - x2 + cos2x+ 2sec2x   b) - 1 sinx cosx- 2(x- 3)+ 12 2
c) 1sin2x+ 2 tan2 x- (x- 3)2 2   d) 2 2 cosx sinx + sec x - x
e) 2 2 2sec x +3 + sinx cosx+ 6x - x

 

There is at least one mistake.
For example, choice (a) should be false.
There is at least one mistake.
For example, choice (b) should be false.
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.
There is at least one mistake.
For example, choice (e) should be true.
Your answers are correct
  1. False.
  2. False.
  3. True.
  4. False.
  5. True.
 

Question 10

 
 
The equation 2xy + y3x = 1 defines y implicitly as a function of x near the point with coordinates (0,1) . What is the value of dy dx- at this point?
a) 2ln2   b) - 1- ln2
c) 1+-2ln2- 2   d) 1- ln2
e) ln2 - 2

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.