# 6-1 nth Root Functions

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```					6-1: nth Root Functions                           Pre Calc I (CP)                                   _______________

Def: Let (n) be an integer with               n2.            x is an nth root of k iff xn = k.

EX 1: _____ is a 4th root of 81 because ___________________________

_____ is a 3rd root of (-8) because ____________________________

 n = odd means 1 root (always)
 n = even means there may be 0, 1, or 2 roots
EX 2:
(a) (n = odd) Find the 3rd root of -8:        (b) (n = even) Find all second roots of 16:
_______                                       ___________
rd
Find the 3 root of 0: _______________         Find all second roots of 0:
_____________
rd
Find the 3 root of 8: ______________          Find all second roots of -16:
_____________
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Graph y = x and y = 8, y = -8, y = 0. Why Graph y = x2 and y = 16, y = 0, y = -16.
is this true?                                 Why is this true?

Power of A Power Postulate:                   (power to a power, multiply exponents)
For any nonnegative base x and any nonzero real exponents m and n, ( x )  x
m n    mn

1
Def: When       x0    and (n) is an integer with                    n2,             n
xx   n
is the positive nth root of x.

Def: When x<0, and (n) is an odd integer with                                    n  3,     n
x=   the real nth root of x.

EX 3: Evaluate without a calculator:
3
1
 1
(a)   25   2
(b)   16 
4
(c)   3
27      (d)        4
81           (e) 10,0001/2
    

_______             _________                    ________                           ________             ___________
1
Note on notation: This text does not define                              x   n
when (x) is negative. This is why:
1                      2
Suppose you wanted to evaluate                    ( 8)   3
and  8 6 . If this is possible for a negative base,
you should get the same answer:
1
1                                 2
( 8) 3    = _____                   8 6 = ____________ = ________________

1
Therefore, we say          ( 8) 3      doesn’t exist, but         3
8   does exist.

Also, even roots of negatives don’t exist (except 2 = imaginary).
1                        1
EX 4: Graph          f ( x)  x 5       and   g ( x)  x 2 .   Remember your domain:

EX 5: Compare:
1               1                                     1          1
(a) For 0<x<1             3
x _____ x        6
(b) For x>1               3
x _____ x   6

EX 6: The Surface area of a sphere has the equation                                 S  4 r 2 .   Write the formula in terms
of (r).

HW: p. 375 # 1, 4-7, 10-12, 14, 16, 19-25.

6-2: Rational Power Functions                                   Pre Calc I (CP) ______________

Rational Exponent Theorem: For any real number x > 0, and positive integers n and m,
m
m
 1           1
x   x n    x   n xm
n             m n

 
^--- usually take the nth root first because it makes the #s smaller
This is read as the nth root of the mth power of x, or the mth power of the nth root.
5
3
EX 1: Simplify without a calculator                               8

2
_______ is the 3rd root of the 5th power of 8. It is also the 5th power of the 3rd root of 8.

EX 2: Rewrite EX 1 using radical notation

EXPONENT PROPERTIES:
(1) Product of Powers: xm xn = xm+n               (same base, add exponents)
(2) Power of a Power: (xm)n = xmn                    (power of a power, multiply)
(3) Power of a Product (xy)m = xm yn                 (distribute the power)
xm
(4) Quotient of Powers:         =   x mn
xn
m
x   xm
(5) Power of a Quotient:        m                  (distribute the power)
 y  y
(6) Zero Exponent Thm: b0 = 1 (anything to the 0 power except 0 equals 1)
(7) Negative Exponent Property: x-n = 1n
x
m
           1
(8) Rational Exponent Thm. (negative)             x           n
    m
xn

EX 3: Rewrite as a power with a rational exponent:                         7
x9   ______________
3
EX 4: Rewrite as a power:           x4      _______________

EX 5: Evaluate without a calculator:
2                                            1

(a)   64   3
(b)   125       3
(c)   3
27

3
3

 16        4
EX 6: Simplify without a calculator:       
 81 

EX 7: The Surface Area of a cube with side (s) is SA = 6s2. (V = s3).
Find the surface area of a cube with volume 128 cm3.

HW: p. 380-382 # 1-14, 16, 20, 22, 23.

6-3: Logarithm Functions                Pre Calc I (CP) _______________

 The inverse of y = bx (switch x,y) is     x  by .     To solve this for (y), rewrite as a
logarithm:

Def:   If b >0 and b1, then (y) is the logarithm of (x) to base b written y  logb x iff by
= x.

Common logs = base 10
Natural logs = base (e)

EX 1: Evaluate without a calculator:

(a) log 1000           (b)   log 2 8    (c) log9 27             (d) log2 64

4
EX 2: Solve for (x):            10 x  6                    EX 3: log (x) = 2.4787

EX 4: Graph y  log 2 x without a calculator.
Its the inverse of y = 2x, so use your calculator to get ordered pairs for y = 2x and switch.
6

4

2

-5                  5

-2

-4

-6

Asymptotes of y = log2 x _____________________

Domain: ____________

Range: _____________

HW: p. 387-388 # 1-7, 9-18, 21-24.

6-4: e and Natural Logarithms                   Pre Calc I (CP) ________________
n

EX 1: Consider 11  1  as n approaches infinity. Use your calculator....
      
    n

5
The number e = Euler’s constant.     e  ______________________________

(e) is used in continuous change models – for example, interest that is compounded
continuously.

Continuously compounded interest: principal P invested at rate (r) for time (t) years,
A(t) = Pert . This continuous change model applies to things besides money.

EX 2: Find the annual yield of money invested at 4% compounded continuously for 6
years. (Recall, annual yield is the % interest of putting 1 dollar in an account earning
simple interest for 1 year).

EX 3:   If \$10,000 is put into a 5-year CD at 7.35% interest compounded continuously.

(a) What is the balance at the end of the period?

(b) How does this compare with the balance if the interest were compounded
annually?

If y = ex, the inverse would be x = ey. Write this as y = ln (x)

ln (x) is the natural logarithm of (x) which is   log e x   (a log with base (e)).

6
EX 4: Approximate with a calculator:

(a)   ln 22   = __________ (b)         log e 17 =   _________        (c)   ln e =   ________

(d) ln 0 = _________              (e) ln 1 = ____________

EX 5: Solve for x:           ln x  4

EX 6: Solve:      e x  22

HW: p. 394-396 # 1, 4-11, 13, 18-26.

6-5: Properties of Logarithms                        Pre Calc I (CP) _______________

Properties of logs operate just like exponents – because logs are exponents.

For every base b,     log b 1  0

For every base b and any real number n,                logb bn  n

For any base b and for any positive real numbers x, y,               logb ( xy )  logb x  logb y

7
x
For any base b and for any positive real numbers x, y,          log b    log b x  log b y
 y

For any base (b) and positive real number (x) and any real number (p):

logb x p  p logb x
EX 1: Simplify without a calculator:

(a) log 6 2  log 6 3 (b)   log5 200  log 5 8                    (c)
1
log85  log17  log 400
2

EX 2: Rewrite without logs:
1
(a) ln x  ln a                              (b)    log x  log a  4log b
3

EX 3: Rewrite without exponents:       A  Pert

HW: p. 401-402 # 1, 3, 6-11, 13-16, 19-25.

8
PRECALC I – SECTION 6.5 WORKSHEET      USING PROPERTIES OF
LOGS

EXPRESS THE FOLLOWING IN TERMS OF a, b, and c,
GIVEN: log 2 = a  log 3 = b  log 5 = c

1. log 4 =

2.log 25 =

3. log 10 =

4. log 6 =

5.log 15 =

9
6. log       =
5

7. log   4
3   =

8. log   3
25   =

9
2
9. log   3
=
5

10. log          6   =

10
6-6: Solving Exponential Equations            Pre Calc I (CP) ___________

EX 1: Solve 3x = 57               Take logs of both sides: (any base)
Method 1: (graph)                            Method 2:

EX 2: Solve    32x = 50

log t a
CHANGE OF BASE RULE:            logb a = log b
t
(To use common logs (t=10), value = "log of the number" over "log of the base")

EX 3: Find log17 34                           EX 4:      log8 256

EX 5: Solve for (x):      3 log 2 5  2 log 2 6  log 2 (5x)

EX 6: \$4,500 is invested at 8% compounded continuously. How long will it take to reach
11
\$11,000?

EX 7: The population of the US reached 250 million in 1990 and was growing at about
1% per year. Use the continuous change model to find out, if the growth rate continued
unchanged, when the population will reach 300 million.

HW: p. 406-408 # 1, 2(c), 3-10, 11 (a-c), 13-22.

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

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SIMPLIFY EACH OF THE FOLLOWING WITHOUT A CALCULATOR
4                         4
1.   32 5                 2.   27 3

 12                     23
 1                       8 
3.                      4.    
 25                      27 

5.   log 6 1              6.   ln e10

7.   log 1  25
5
8.   log 2 8

 1 
10.  x6 y15               
1

9.   log 9  
3

 27 

13
EXPRESS EACH OF THE GIVEN LOGARITHMS AS THE SUM AND/OR
DIFFERENCE OF SIMPLIER LOGS, WITHOUT ANY RADICALS OR EXPONENTS

 x
11. log  ab3c                                        12.       ln  2 
 y 
   

EXPRESS EACH OF THE FOLLOWING AS A SINGLE LOGARITHM WITH A
COEFFICIENT OF ONE

13.   log x  log y  3log z                           14.       2  ln p  ln 3

15. Express this equation without logs:                  log x  log y  log 5  log w  2log z

16. Determine the value of x (correct to the nearest thousandth):                            ln x  2

4
17. Determine the exact value of x:                 log 27 x 
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 x         21.       hx  log 2 x       22. rx  0.2x
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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.

15
SOLVE EACH OF THE EQUATIONS

27.   log 6 9 x  4  log 6 19   28.     4x  8

x
 1 
29.     25    x2
                    30.   92 x 1  27 x  2
 25 

31.   log x  4 log 5  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.

33.   g x   log 3 x             34. r x   4 x

SOLVE EACH OF THE GIVEN EQUATIONS, CORREST TO THE NEAREST
THOUSANDTH.
35. 6 x  2.414         36. 151.08x 1  400

6e x  800
2
37.

16
EVALUATE EACH OF THE FOLLOWING WITHOUT A CALCULATOR, GIVEN
THAT:
log8 5  x AND log8 12  y

38.   log 8 60                          39.   log8 144

40.   log8 2.4                          41.   log8 300

EVALUATE EACH OF THE FOLLOWING

42.   log 6 344                         43.   log15  20 4 

44. The altitude above sea level h (in feet) as a function of barometric pressure p (in
lb/in2) can
be approximated by the formula h  28,300 ln  p  . What is the altitude of Santa Fe,
    
 14.7 
New Mexico, if the average barometric pressure there is about 11.5 lb/sec2?

17

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