6.1: nth Root Functions CP Pre‐Calc I A. B. Evaluate each of the following without a calculator 1
1. 2. 64 3 4. 3. 3 −125 Rewrite each with a radical sign 1
5. x2 4 Halldorson 1
36 2 4
16 6. ( )
1
m6 C. Rewrite without a radical sign 7. 3w2 8. 3
w D. 1
4
1
3
Graph f ( x ) = x and g ( x) = x on the same window. For what values of x is: 9. f(x)<g(x) 10. f(x)>g(x) 11. f(x)=g(x) CP Pre‐Calculus I A. 6.2: Rational Power Functions Evaluate each of the following. Show all work. 1. 3 0.027 2. Halldorson 3
⎛ 4 ⎞2
⎜ ⎟ ⎝ 25 ⎠
3. B. −
3
( 6)
1
− 8
Solve for x 5. 2 x = 16 7. 2
72 =
4. 1
⎛ 1 ⎞ 6
⎜ ⎟ ⎝ 64 ⎠
6. x
8. 8 x = 16 3 = 27−2 x LOGARITHMS PRACTICE WORKSHEET
NO CALCULATORS!!!
Find each logarithm:
1. log2 64
4. log 104
2. log4 2
5. log1/3
3. log4
1
16
1
81
6. log1/4 64
Find the value of x in each equation:
7. logx 625 = 4
13. log (x2 + 9x) = 1
8. log25 x = ½
14. log (4x – 4) = 2
9. log1/2 x = -6
15. log5 x = 3 log5 7
10. logx .1 = -1
16. log2 x = ½ log2 81
11. logx 125 = -3
17. log5 x = 2 log5 7
12. log√3 27 = x
18. log10 x =
1
(2 log 4 + 2 log 2)
6
Express each logarithm as the sum or difference of simpler logs:
19. log2 (xy) =
20. log2 (abc) =
21. loga 2x1/2 =
22. loga (bc)2 =
23. loga
bc =
24. logb (
x
)=
y
Express each as a single logarithm with a coefficient of 1:
25. loga x + loga y – loga z =
26. 2 loga x – ½ loga y =
27. 2(loga z – loga 3) =
Simplify (WITHOUT A CALCULATOR):
28. log6 2 + log6 3
29. log5 200 – log5 8
30. log 85 – log 17 + ½ log 400
8Name ________________________________
Logarithm Practice Worksheet
Simplify WITHOUT a calculator:
2
2.
1. 1000 3
4.
(4 )
3
3
9⋅6 9
3. log3 9
3
5. log9 3
6. log4 8
8. log10 .0001
9.
10. log2 8
11. log8 ½
12. log9 27
13. log8 4 2
14. log4 x = 3
15. logx 8 =
7. 32
−
2
5
( 6 )( 6 )
2 −π
1+π
3
4
Suppose f(x) = 1 – x3 and g(x) = 2x – 3. Find the following:
16. f(g(1))
17. g(f(-1))
18. f-1(9)
19. g-1(0)
20. f(f-1(4))
21. g-1(f-1(0))
If log 5 = a and log 4 = b, express the logarithm in terms of a and b.
Ex. log 16 = log 42 = 2log 4 = 2b
4
16
23. log 20
24. log 1.25
25. log 2
26. log 10
27. log
1
25
28. log 250
29. log 10
30. log
3
22. log
50
Let x = log 2, y = log 3, and z = log 10. Express each logarithm in terms of x, y, and z.
31. log 6
32. log 9
33. log 5
34. log 18
35. log 1.5
36. log .75
37. log 80
38. log
39. log .0006
40. log
9
2
30
41. log 1200
Let log a = 2, log b = 3, and log c = 4. Evaluate each logarithm.
42. log a2b
45. log
6
43.
a 2b3c
3
bc 2
46. log
a 3b 2
c4
Simplify to help you solve each equation.
48. log3 x + log3
1
=0
5
49. 2log4 y = 3
50. If log8 5 = a, then log8
1
= _________
5
44. log
a
bc
47. log
a2
bc 2
PRECALC I – SECTION 6.5 WORKSHEET
PROPERTIES OF LOGS
USING
EXPRESS THE FOLLOWING IN TERMS OF a, b, and c,
GIVEN: log 2 = a
1. log 4 =
2.log 25 =
3. log 10 =
4. log 6 =
5.log 15 =
6. log
9
5
=
7. log
4
3
8. log
3
25
=
=
log 3 = b
log 5 = c
9. log
3
10. log
2
5
=
6
=
Chapter 6 Guided Notes
Halldorson
2009-2010
PRECALC I
CHAPTER 6 STUDY GUIDE
1. Know how to simplify exponential and logarithmic expressions with and without a calculator. (#1-8, 17-22,
25, 26)
2. Know the properties of logs. (#37-43, 47, 48)
3. Be able to solve equations. (#13-16)
4. Know what the graphs exponential and logarithmic functions look like. Know the characteristics, like
domain, range, asymptotes. (#30, 32-34, 69, 70)
5. Be able to give formulas in terms of another variable and then substitute in values. (#55a, 56, 59, 62)
6. Be able to solve compound interest problems and population problems. (#58, 60)
PRECALC I – CHAPTER 6 REVIEW WORKSHEET SIMPLIFY EACH OF THE FOLLOWING WITHOUT A CALCULATOR 4
1. 32 5 − 12
2. 27 3 ⎛ 1 ⎞
3. ⎜ ⎟
⎝ 25 ⎠
5. log 6 1 6. ln e10 8. log
9. log 9 ⎜
4
⎛ 8 ⎞
4. ⎜ ⎟
⎝ 27 ⎠
−23
7. log( 1 ) 25 5
2
8 (
6 15
10. x y
)
( 13 )
1
⎛ 1 ⎞
⎟ ⎝ 27 ⎠
Chapter 6 Guided Notes
Halldorson
2009-2010
EXPRESS EACH OF THE GIVEN LOGARITHMS AS THE SUM AND/OR DIFFERENCE OF SIMPLIER LOGS, WITHOUT ANY RADICALS OR EXPONENTS (
)
3
11. log ab c ⎛ x⎞
⎜ y 2 ⎟⎟
⎝
⎠
12. ln ⎜
EXPRESS EACH OF THE FOLLOWING AS A SINGLE LOGARITHM WITH A COEFFICIENT OF ONE 13. log x + log y − 3 log z 14. 2 ( ln p − ln 3) 15. Express this equation without logs: log x − log y = log 5 + log w − 2 log z 16. Determine the value of x (correct to the nearest thousandth): ln x = 2 17. Determine the exact value of x: log 27 x =
−4
3
18. Determine the exact value of m: log m 8 = 6 MATCH EACH EQUATION WITH ITS GRAPH 19. f ( x ) = − ln x 20. g ( x ) = 2 21. h( x ) = log 2 x 22. r ( x ) = (0.2 ) x
x
4
4
4
4
2
2
2
2
5
5
-2
-2
a.
-2
b.
-4
-2
c. 2
d. Chapter 6 Guided Notes
Halldorson
2009-2010
23. Matt invested $1,000.00 in an account paying 7.5% interest. Determine his balance after four years if interest is: a. compounded monthly b. compounded continuously 24. Nicole has invested $5000 in an account paying 6.25% interest compounded continuously. How long before her investment doubles in value? 25. In 1995 the population of Mexico was estimated at 94,800,000 with an annual growth rate of 1.85%. Assuming continuous growth at this rate, estimate the population of Mexico in the year 2010. 26. The population of Peru in 1998 was estimated at 18,4000,000 with an annual average growth rate of 2.7%. Assuming continuous growth at this rate, in what year would the population reach 30,000,000. SOLVE EACH OF THE EQUATIONS 27. log6 (9 x + 4) = log6 19 x
4 = 8 x
28. 29. 25
31. log x − 4 log 5 = −2 x+2
⎛ 1 ⎞
=⎜ ⎟ ⎝ 25 ⎠
30. 9 2 x +1 = 27 x + 2 32. ln ( x − 1) − ln ( x + 1) = −2 FOR EACH OF THE GIVN FUNCTIONS: a). DRAW AN ACCURATE GRAPH, b). STATE THE DOMAIN, c). STATE THE RANGE, AND d). STATE THE EQUATION OF ANY ASYMPTOTE(S). 33. g ( x ) = log 3 x 34. r ( x ) = 4 x
-10
10
10
8
8
6
6
4
4
2
2
-5
5
10
-10
-5
5
-2
-2
-4
-4
-6
-6
-8
3
-8
10
Chapter 6 Guided Notes
Halldorson
2009-2010
SOLVE EACH OF THE GIVEN EQUATIONS, CORREST TO THE NEAREST THOUSANDTH. 35. 6 x = 2.414 36. 15(1.08 )
x −1
= 400 37. 6e
x2
= 800 EVALUATE EACH OF THE FOLLOWING WITHOUT A CALCULATOR, GIVEN THAT: log8 5 = x AND log8 12 = y 38. log8 60 39. log8 144 41. log8 300 40. log8 2.4 EVALUATE EACH OF THE FOLLOWING (round your answer to the nearest thousandth). 42. log 6 344 (
)
4
43. log15 20π 44. The altitude above sea level h (in feet) as a function of barometric pressure p (in lb/in2) can be approximated by the ⎛ p ⎞
⎟ . What is the altitude of Santa Fe, New Mexico, if the average barometric pressure ⎝ 14.7 ⎠
formula h = −28,300 ln ⎜
there is about 11.5 lb/sec2? 4