Download Logarithm Review - davis.k12.ut.us

1050 Exam 3 Review
x) 2 x − 3 and g ( x=
1. Let f (=
) 2 x + 3 . Find each of the following:
a. ( f  g )( x)
Domain: _______________
2
b.
( f  g )(−3)
2. If f ( x) =
2
2x + 1
and g ( x) =
find ( f  g )( x) and the domain.
x+3
x +1
3. Show that f (x) and g (x) are inverses by finding ( f  g )( x) and ( g  f )( x) .
f ( x) = 2 − x 3
and
g ( x) = 3 2 − x
4. a. Find the inverse of the function:
f ( x) = x 2 + 2
x≥0
b. Graph both f (x) and f
coordinate plane.
−1
( x) on the same
y
x
f
−1
( x) =_______________
Domain of f
−1
( x) _______________
5. a. Find the inverse of the function:
f ( x) = x 3 − 1
b. Graph both f (x) and f
coordinate plane.
−1
( x) on the same
y
x
f
−1
( x) =_______________
Evaluate each log using the change of base.
7. log 8 15 = ___________
6. log 25 80 = _________
Evaluate each log using the properties of logs (without a calculator).
8. log 7 7
9. log 2 16
11. ln e x
10. log 5 1
Use the properties of logs to expand as far as possible.
8x 2
12. log 6 ( yz ) 3
13. log 2 4
yz
14.
ln
5x 3
y
Condense each expression to a single logarithm.
15. 4 log 3 x + 2 log 3 y − 4 log 3 z
16.
1
ln x + 2 ln( x + 1) − ln( x − 2)
3
Graph the following. Give the domain, range, and asymptote of each.
17. f ( x) = 4 x
18. f ( x) = 4 ( x + 2 ) − 3
y
y
Domain:
x
Range:
Asymptote:
Domain:
x
Range:
Asymptote:
20. f ( x) = log 4 ( x + 3) + 1
19. f ( x) = log 4 x
y
Domain:
y
Range:
x
20. graph #2
y
x
Asymptote:
𝑥𝑥+5
20. graph both from #3
𝑓𝑓(𝑥𝑥) = 3𝑥𝑥+5
Domain:
y
Range:
x
x
Solve for x.
22. 3 x +5 − 1 = 6
25. 3 + 4 log 4 x = 10
Asymptote:
23. 2e x + 6 = 38
24. 2 7 x −5 = 16 x
26. 5ln (2x) = 35
27. log 2 ( x + 3) + log 2 ( x − 3) = 4
28. You borrow $3500 at an interest rate of 4.5%.
What will your final balance be at the end of
3 years if you compound
a. Monthly:
b. Continuously:
29. Determine the time necessary for $1,000 to
double if it is invested at 6% interest rate
compounded continuously.
30. A bacteria culture contains 250 bacteria. If the number of bacteria grows to 300 in 5 hours,
find the rate of growth. How long will it take the population to double in size?
Rate = ________________
(don’t round)
31. Solve by Gaussian Elimination
a. x + y + z = 1
3 x − y − z = 11
x + 3y = 0
b. x + 4 y − 2 z = −3
2 x − y + 5 z = 12
length of time to double = ____________________
(round to 2 decimal places)
32. Find the value of the determinant
6 −2
−4 −2
33. Solve by finding the determinants and using Cramer’s Rule.
x + 2 y − z = −4
x + 4 y − 2 z = −6
2x + 3y + z = 3
a. A − 3B
0
1 − 2 

1  , C = 4 3  and find:
0 − 1 
0 
b. 3 A + 2 B
c. BC
d. CA
4
 5

34. For the matrices A =−
 4 −4
 1
2
4
4
4


0  , B = 3 2
1 − 2
0 
35. Solve the equation using Matrix Algebra.
2x − 6 y =
0
x − 2y =
2
Solve the non-linear systems by substitution or elimination.
x2 − y =
7
x 2 + y 2 = 13
37.
36.
2x2 + y 2 =
17
x2 − y2 = 5
Review Answers
1. a. 8 x 2 + 24 x + 15
4.
f
−1
D: R b. 15
( x) = x − 2
−1
5. f
D:x≥ 2
2.
x+5
3x + 5
( x) = 3 x + 1
D : R, x ≠ −1,
6. 1.361
−5
3
3. Show every step!
7. 1.302
8. 1
9. 4
10. 0
11. x
y
12. 3(log 6 y + log 6 z )
y
13. 3 + 2 log 2 x − log 2 y − 4 log 2 z
x
x
14.
1
ln 5 + 3 ln x − ln y
2
15. log 3
17. Domain: (−∞, ∞) 18 Domain: (−∞, ∞)
Range: (0, ∞)
Range: (−3, ∞)
Asymptote: y = 0
Asymptote: y = −3
x
24. x = 5
3
23. x = 2.773
26. x = 548.317
27. x = 5 , -5 28. a. $4004.87 b. $4005.88
30. Rate = .036464311 time = 19.01 hours
33. (−2, 1, 4)
y
x
22. x = −3.229
x
25. x = 11.314
29. t = 11.552 years
31. a. (3, −1, −1) b. (−2z + 5, z – 2, z)
34.a. −7
−8 4  b.  23 20 12 
 −6 −8 2 
 −13 −10 −3




 5
2 0 
 −2
8
0 
37.
x ( x + 1) 2
x−2
y
x
35. (6, 2) 36. (3, 2, 1)
3
16. ln
19. Domain: (0, ∞)
20. Domain: (−3, ∞)
Range: (−∞, ∞)
Range: (−∞, ∞)
Asymptote: x = 0
Asymptote: x = -3
y
y
x4 y2
z4
c.
(2,−3) (−2,−3)
(2 2 ,1) (−2 2 ,1)
 20 4  d. Not possible
Wrong Orders
 11 − 1


− 7 − 8
32. −20