Alg 3 Functions
1
Algebra 3 Assignment Sheet
Functions, Fog, Gof, Inverse, Logs
(1) Assignment # 1 – Functions, Domains
(2) Assignment # 2 – Composition of Functions
(3) Assignment # 3 – Inverse Functions
(4) Review Sheet
(5) Quiz
(6) Assignment # 4 – Exponential Equations
(7) Assignment # 5 – Logarithms
(8) Assignment # 6 – Laws of Logarithms
(9) Assignment # 7 – Laws of Logarithms
(10) Assignment # 8 – Calculator Logarithms
(11) Review Sheet
(12) TEST
Alg 3 Functions
2
Functions
I Vocabulary
a) Function: The set of (x, y) pairs such that____________________________________
____________________________________________________________
b) domain:
c) range:
Vertical Line Test
Y/N
Domain
a) (2,4) (8,1) (8,8)
_____
______________
b) (1,-2) (3,4) (5,-2)
_____
______________
c) y = x 2 + 5x + 6
_____
______________
d) y = |x|
_____
______________
e) y 2 = 8
_____
______________
f) y = x
_____
______________
_____
______________
_____
______________
II Functions ?
A.
g) y = - ( x + 2 )
h) f (x) =
2
2x
x −4
2
f(x) is the same as y
Alg 3 Functions
III Restrictions
3
1) Denominator ≠ 0
2) for even roots:
5+ x
,
x
2 - 3x
,
x -5
x≠
x≠
x
x + 5x - 6
2
x≠
negative number
Find the domain of the following please.
a) f (x) =
x−2
d) y = x 2 − x − 12
b) f (x) =
c) f (x) =
2x + 5
e) y =
x 2 − 2x − 63
x 2 + 8x + 15
x
x−2
Alg 3 Functions
IV Evaluating functions for y when given a value of x
Evaluating: Substituting in numbers
y = x2 + 2x + 3
f ( x) = x 2 + 2 x + 3
f (2) = 2 + 2 ( 2 ) + 3 = 11
2
4
y = f(x)
f ( x) = x 2 + 2 x + 3
f ( −1) = ( −1) + 2 ( −1) + 3 = 2
2
1)
f (x) = x 2 + 2x + 3
f(-1) =
f(0) =
 1
f  =
2
2)
g(x) = x 2 − 2x + 3
Find g(x + 2)
3) If f(x) = x 2 − 3 x + 1
a) find f(x + 2)
c) find f(g(x))
g(x) = x - 5
b) find 3f( x) • g(2x)
Alg 3 Functions
5
4
3
(1) f ( x ) = 3x − 16x −
Algebra 3 Assignment # 1
2
2
7x − 28x − 13 , g ( x ) = x + 2x − 4 . Find each of the following.
(
)
(a) g ( 4 )
(c) f g ( −3)
(b) f ( 6 )
(d) The remainder when f ( x ) is
divided by ( x − i )
(2) Determine whether each of the following defines y as a function of x. If it is a function, find the domain
please.
(a) y = 5x − 4
(b)
2
= x + 2x − 3
y
(c) x
2
+ y
2
= 1
(f) y =
(g) y =
(h) y =
5 − 4x
2
x − 5x − 6
x
2
x + 1
(d) y =
3x + 5
2x − 3
(i) y =
3
(e) y =
x + 8
x 3 − 7x + 6
(j) y =
5 − 2x
3x + 4
3
2
x − 9x + 23x − 15
(3) Let the function f be defined by f ( x ) = x + 1 . If 2 f ( ϑ ) = 20 , find f ( 3ϑ ) .
(4) Let the function f be defined by f ( x ) = x + c , where c is constant. If f ( 2 ) = 10 , find the value of the
constant c.
Alg 3 Functions
6
Answers
(1) (a) 20
(c) 27
(b) –1
(d) –3 – 12i
(f) x ≤ 5
(2) (a) Real
4
(b) Not a function
(g) x ≤ − 1 or x ≥ 6
(c) Not a function
(h) Real
(d) x ≠ 3
(i) − 4 ≤ x ≤ 5
(e) x ≠ 1 , 2 , − 3
(j) 1 ≤ x ≤ 3 or x ≥ 5
2
(3) 28
3
(4) 8
4
Alg 3 Functions
7
Composition of Functions
Functions can be added, subtracted, multiplied and divided.
EXAMPLES
1. If f (x) = x 2 − 3 and g(x) =
a) f(x) + g(x)
d) f(3)
Notes:
g(3)
1
find
x
b) f(x) g(x)
e) f(3) ÷ g(3)
Let f (x) = x and g(x) = x 3 − 3
f g (x) = f(g(x))
Use composition to find the following:
1
If f (x) = x 2 − 3 and g(x) = find
x
b) g(x) f (x)
a) f (x) g(x)
If f (x) = 2x − 1,
g(x)=x 2 and h(x)=
1
3x
find f g h(x)
c) f(x) ÷ g(x)
Alg 3 Functions
x−6
x+1
If f (x) =
and g(x)=
x−4
x+5
8
find f g
DOMAIN:
f ( x ) = x 2 + 8x + 15 , and g ( x ) = x + 3.
If f (x) = x and g(x)=
1
x −2
2
find the domain of g f
Find all values of x such that f ( g ( x ) ) = g ( f ( x ) ) if
Alg 3 Functions
9
Algebra 3 Assignment # 2
Composition of Functions
(1) Find each of the following numbers, given the functions below.
f (x ) =
h (x ) = x 2 + 1
2x − 1 ; g (x ) = 2x 2 − x − 2 ;
(a) f (h (2 ))
(b) g (h (1))
(c) g (f (25 ))
(d) h (f (g (3)))
(2) Find f (g (x )) and g (f (x )) for each of the following please.
(a)
f (x ) = 3x
2
+ 2x − 1
g(x ) = 4x − 5
(3) Find g (x ) if f (x ) =
2x + 5
4x − 3
(b)
x+2
g(x ) =
3x − 1
f (x ) =
x +9
3x − 1
, and f (g(x )) =
.
12x − 11
2x + 5
(4) Find f (f (x )) if f (x ) =
2x + 3
3x − 2
2
(5) Find all values of x such that f (g (x )) = g (f (x )) if f (x ) = 2x − 3x + 2 , and g(x ) = 3x − 2.
Alg 3 Functions
10
Answers
(1)
(a) 3
(b) 4
(c) 89
(d) 26
(2) (a) f (g (x )) = 48x
2
− 112x + 64
g (f (x )) = 12x 2 + 8x − 9
(3) g (x ) =
x + 2
2x − 3
(4) f (f (x )) = x
(5) x = 1
(b) f (g (x )) =
17x − 1
− 5x + 11
g(f (x )) =
10x − 1
2x + 18
Alg 3 Functions
11
Inverse functions
(f
and
f −1 )
f −1 is not
1
f
NOTES:
A.
then
If f(x) and g(x) are inverses of each other
f(g(x)) = x
and
g(f(x)) = x
(they must be one to one to be inverses)
Ex. If f(x) = 2x - 3
g(x) =
x+3
2
g(x)
 x + 3
f

 2 
=
f(x)
g [ 2x − 3] =
 x + 3
2
−3 = x + 3 – 3 = x
 2 
( 2x − 3) + 3
2
=
2x
2
=
x
Alg 3 Functions
12
If f (x) = x 2 and g(x) = x , Prove f(x) and g(x) are inverses of each other
B.
Finding inverses
• Switch the domain “x” and the range “y”
• Then solve for “y”
EX.
f(x) = 2x - 5
y = 2x – 5
solve
Ex. Find f −1 if
f ( x ) = 3x − 2
switches to
x = 2y – 5
x +5
this is the inverse
y=
2
Domain restrictions become:
Alg 3 Functions
V Practice
13
−1
Find f (x) for each of the following:
a) f ( x ) = x + 1
3
b) f ( x ) = ( x + 2 ) − 3
2
Find
c) f ( x ) =
5x − 1
2
f (g ( x ))
and
e)
g ( f ( x ) ) for each of the following:
f ( x ) = 4x − 3
f)
g(x) = x + 6
f (x) = x + 3
g ( x ) = x 2 + 2x − 4
Find each of the following numbers, given the functions below:
f (x) = x + 4
,
g ( x ) = 3x 2 − x + 2,
h ( x ) = x2 − 2
g) f ( h ( 4 ) )
h) g ( h ( −2 ) )
j) h ( f ( 0 ) )
Alg 3 Functions
14
Algebra 3 Assignment # 3
Inverse Functions
(1) Find f
−1
(x ) for each of the following please.
(a) f (x ) = 5x + 3
(e) f (x ) =
5x − 7
(b) f (x ) =
4
x
(f) f (x ) = −
(c) f (x ) =
3x + 2
5x − 2
(g) f (x ) = 3 4x + 5
(d) f (x ) =
7x + 2
2x − 7
(h) f (x ) = − 3 5x + 8 − 2
(2) f (x ) =
(3) f (x ) =
4x + 5 + 2
6x + 5
, g (f (x )) = x . Find g (x ) .
2x + 3
x − 2 . Find f −1 (x ) , and sketch a graph of f (x ) and f −1 (x ) on the same set of axes.
Alg 3 Functions
15
Answers
(1)
(a) f
(b) f
(c) f
(d) f
(2) g (x ) =
−1
(x ) = x − 3
5
−1
(x ) = 4
x
−1
(x ) =
−1
2x + 2
5x − 3
(x ) = 7 x + 2
2x − 7
− 3x + 5
2x − 6
(e) f
(f) f
(g) f
(h) f
−1
−1
x2 + 7
(x ) =
5
x 2 − 4x − 1
(x ) =
4
−1
−1
(x ) =
x3 − 5
4
(x ) =
(
x + 2 )3 +
−
5
8
Alg 3 Functions
16
Algebra 3 Review Worksheet
(1) Find each of the following numbers, given the functions below.
f (x ) = x 2 − 2x ; g (x ) = 3x ; h (x ) = x + 1
(a) f (h (3))
(b) g (h (0 ))
(c) f (h (g (8 )))
(d) g (f (h (8 )))
(2) Find the domain of each of the following functions please.
(a) f ( x ) =
(c) f ( x ) =
24x 2 − 29x − 4
5x + 2
3
2
x − 4x + x + 6
(b) f ( x ) =
(d) f ( x ) =
x − 1
x2 − 9
( x + 3)( x − 1)2
( x − 5)
(3) Find f (g (x )) and g (f (x )) for each of the following.
f (x ) = 2x + 1
(a)
(4) Find f
g (x ) = x 2 − 3
−1
f (x ) =
2x + 3
3x − 2
g (x ) =
x +1
2x − 1
(b)
(x ) for each of the following.
(a) f (x ) = 5x − 7
(c) f (x ) =
3x + 2
2x − 5
(b) f (x ) = 3 2 x + 5 − 4
(d) f (x ) = −
5x + 1
(5) Find all values of x for which f (g (x )) = g (f (x )) if f (x ) = x − 5 and g(x ) = 2x
(6) Find g (x ) , if f (x ) =
2x + 1
6x − 1
and f (g(x )) =
.
x + 2
3x + 1
2
− 4x + 3 .
Alg 3 Functions
17
Answers
(1)
(2)
(a) 0
(b) 3
(c) 15
(d) 9
(a) x ≤ − 1 or x ≥ 4
(b) x ≥ 1 and x ≠ 3
(c) x ≠ − 1 , 2 , 3
(d) x ≤ − 3 or x = 1 or x > 5
8
3
(a) f (g(x )) = 2x
(3)
2
− 5 , g(f (x )) = 4x 2 + 4x − 2
8x − 1
5x + 1
, g(f (x )) =
−x + 5
x + 8
x+7
(
x + 4 )3 − 5
−1
−1
(b) f (x ) =
(a) f (x ) =
2
5
(b) f (g (x )) =
(4)
(c) f
(5)
15
4
−1
(x ) =
5x + 2
2x − 3
(d) f
−1
(x ) =
x2 − 1
5
(6) g (x ) = 3x − 1
Alg 3 Functions
18
Exponents
I Review
bxby =
( )
b0 =
b− x =
b 3b
3
bx
( )
=
b
bx = by ,
II If
y
then
x=y
3
bx
=
by
=
bxcx =
2
=
If
bx = a x ,
x
Ex. 3
x+2
=3
5
x+2 = 5
x=3
1) 2 x −1 = 32
2
2) 2 x = 216
3) 82x +1 = 64
4)
1
2x
= 64
5) 27 x +1 =
6) 8x
2
−x
1
9
= 4x
2
+5
b
3
b
2
then
2 2
Ex.   =  
3 3
x=4
=
4
b=a
Alg 3 Functions
7. 3x + 3x + 3x =
19
1
729
8. 5 ⋅ 4x + 4 x = 96
9. 3x + 3x + 3x + 3x +1 = 54
III
Sketching
y = 2x
x
y
-3
-2
-1
0
1
2
3
y = 2− x
x
-3
-2
-1
0
1
2
3
y
1) NO negative bases
2) (0,1) on every graph
3) b ≠ 1
Alg 3 Functions
20
Algebra 3 Assignment # 4
Exponential Equations
(1) Solve for x please.
(a) 4
(c) 2
(e)
x +1
2x − 1
( 14 )
•
= 8x
(b) 5
= 8x − 7
(d) 9
8 2 x − 1 = 161 − x
(f)
−3
(g) x 2 = 27
8
(i) 27
2x
− 10 • 27 x + 9 = 0
x+3
1
= 25
x 2 − 2x
( x2 − 1 )
(h) 2
2x
x
4
3
2+ 1
= 16
− 13 • 2 x − 48 = 0
x
x
(j) 16 − 10 • 4 + 16 = 0
(2) Sketch a graph of each of the following.
(a) y = 3
= 27 x
(b) y = 3
−x
Alg 3 Functions
21
Answers
(1)
(a) 2
(b) −5
(c) 20
(d) −1 , −3
(e)
(f) ± 3
9
10
(g) 4
(h) 4
9
(i) 0 , 2
3
(j) 1 , 3
2
2
Alg 3 Functions
EXPONENTIAL EQUATIONS EXTRA
22
(1) Evaluate each of the following numbers please.
(a) 9
3
2
i 49
2
(b)
−2
2
−1
2
+ 2
(c) 2
2 2
2
2
(d)
−1
i4
2
−3
− 2
−1
−2
− 3
(2) Solve each of the following equations please.
(a) x
(b) x
(c) 8
2
3
= 25
−3
5
( )
(x
(
1
=
8
2x + 3
(d) 8 i
(e)
2
+ 7
)
3
5
=8
2
(f) x − 6x + 9
= 16
1 2x + 1
4
x +4
= 16
(g) 8
x−3
2x
3x
(h) 4
(3) Sketch a graph of each of the following on the same set of axes.
y=2
x
and y = log 2 ( x )
)
3
4
= 27
x
− 6i8 +8 = 0
− 9i4
3x
2
+ 8 = 0
Alg 3 Functions
23
Answers
(1)
(a) 27
(c) 1
(b) 17
(d) 6
(a) 125
(e) ±5
(b) 32
(f) 12 , –6
7
2
(2)
(c) 7
(g) 1 , 2
(d) 13
(h) 0 , 1
2
8
3
3
Alg 3 Functions
LOGS
24
Logs are the inverse of exponential functions.
If y = 2x + 3 to find inverse
x = 2y + 3
x −3
=y
2
If y = 2
x
to find inverse x = 2
How do you solve for y?
Logarithms are exponents and follow exponent rules.
EXPONENTIAL FORM
power
bp = n
base
33 = 27
1
5−2 =
25
3
2 =8
4x = 1
RULES
number
LOGARITHM FORM
number
log b n = p
base
log 3 27 = 3
1
log 5
= −2
25
log 2 8 = 3
log 4 1 =
power
y
Alg 3 Functions
EXAMPLES
25
log 3 81 = x
log8 1 = c
log 2 4 = y
log 27 9 = d
log 5 25 = z
log 9 ( log 3 x ) =
log8 16 = p
log 6 6 = a
log125 x = −
1
2
2
3
warm-up before next lesson
1)
2
-2
-2
5 − 2
4) log 2
3
-4
8
=
27
7) log .001 10 = y
2) 8 i
()
1
4
5) logb
2x + 1
= 16
1
3
=27 2
x−3
3x
3) 4
− 9i4
+ 8 = 0
1
)=x
27 81
6) log 3 (log 1
4
3x
2
Alg 3 Functions
26
Algebra 3 Assignment # 5
Definition of the logarithm
Solve for x please.
( e )=x
(1)
log4 (64) = x
(10)
5ln
(2)
log6 (x ) = 2
(11)
1 ) = −2
log x ( 25
(3)
log x (9) = 2
(12)
log36 (216) = x
(4)
log3 (x ) = − 2
(13)
log 4 (x ) = − 23
(5)
log25 (125) = x
(14)
log 8 (4 2 ) = x
(6)
log8 (x ) =
(15)
log x (6) = − 12
(7)
log27 (81) = x
(16) x =
(8)
log 7 ( 7 ) = x
(17)
log 4 ( log 2 (x ) ) =
(9)
log16 (x ) = − 43
(18)
log16 ( log x (9) ) =
2
3
3
2
ln e2
3ln e
4 ( ) + 5 ( )
1
2
1
4
Alg 3 Functions
27
Answers
(1) 3
(10) 10
(2) 36
(11) 5
(3) 3
(12) 3
2
(4) 1
9
(13) 1
8
(5) 3
2
(14) 5
6
(6) 4
(15) 1
36
(7) 4
3
(16) 33
(8) 1
2
(17) 4
(9) 1
8
(18) 3
3
Alg 3 Functions
28
LAWS OF LOGS
Properties of Logs
I LAWS
II Equations
log b mn = log b m + log b n
m
log b = log b m − log b n
n
log b m = log b n
If
then m = n
( log =log )
log b m = n
If
then b n = m
(log = number)
log b m p = plog b m
III Examples
(
)
1)
log 5 2n 2 + 20 = log 5 ( 32 − 5n )
3)
log 6 48 − log 6 w = log 6 4
4)
log 2 3 + log 2 7 = log 2 x
5)
1
log10 m = log10 81
2
6)
1
1
log 7 m = log 7 64 + log 7 121
3
2
Alg 3 Functions
29
7)
log10 ( m + 3) − log10 m = log10 4
8)
1
2log 6 4 − log 6 8 = log 6 x
3
9)
log 4 ( x + 3) + log 4 ( x − 3) = 2
10)
log 2 ( y + 2) − 1 = log 2 ( y − 2 )
11)
log 3 x − log 3 5 = log 7 7
12)
log 2 ( x + 1) + log 2 (3 x − 1) = log 3 243
Alg 3 Functions
30
Algebra 3 Assignment # 6
Logarithmic Equations
(1) Evaluate each of the following please.
log (6)
(a) 7 7
(c) 4
log8 (27)
log (36)
(b) 5 25
log 4 (25)
+ 8
(d) e
(2) Solve for x please.
2 ln(8) − 3ln(4)
(
)
(a)
log 3(2x + 1) = log 3(3x − 6)
(b)
log10 x 2 + 9x = 1
(c)
log 5 ( x ) = 4log 5 ( 3 )
(d)
log 9 ( x ) = 12 log 9 (144 ) −
(e)
log 3 ( 7 ) + log 3 ( x − 2 ) = log 3 ( 6x )
(f)
ln (15) + ln (14 ) − ln (105) = ln ( x )
(g)
log10 ( x − 1) + log10 ( x + 2 ) = log 7 ( 7 )
(h)
log 3 ( x + 3) + log 3 ( x − 3) = log 3 (16 )
(i)
log 8 ( x + 1) − log 8 ( x ) = log 8 ( 6x + 2 )
(j)
log 3 ( x + 3) + log 3 ( 4x − 1) = log 3 (12 )
(k)
log8 x 2 − x − log8 ( 2x − 5) =
(l)
125x = 8log 4( 9 ) − 3log 9( 4 )
(
)
2
3
1
log 9
3
(8)
Alg 3 Functions
31
Answers
(1)
(2)
(a) 6
(b) 6
(c) 134
(d) 1
(a) 7
(b) −10 , 1
(c) 9
(d) 6
(e) 14
(f) 2
(g) 3
(h) 5
(i) 1
(j) 1
(k) 4 , 5
(l) 2
3
3
Alg 3 Functions
32
Algebra 3 Assignment # 7
Logarithmic Equations
Solve for x please.
(
)
(1)
log 4 x 2 − 1 − log 4 ( 5x − 11) =
(2)
log 6 ( 3x − 5) − log 6 x 2 − 1 = log 6 ( x ) − 1
(3)
ln ( 4x + 1) + ln x 2 + x = ln (19x − 9 )
(4)
log8 3x 2 − 7 − log 8 x 2 − x − 1 =
(5)
ln x 2 + 4 + ln ( 3x − 4 ) = ln (17x − 18)
(6)
2ln 3
log 25
log 2 ( x + 1) + log 2 ( x − 5 ) = e ( ) − 3 9 ( )
(7)
log 3 ( x − 5 ) = log 9 ( x + 7 )
(
(
(
(
)
( log8 ( x ) )
(9) 2
( log 4 ( x ))
(
)
)
(
)
2
3
)
(8) 3
(10)
1
2
2
2
)
− log 8 ( x ) − 2 = 0
+ 5log 4 ( x ) = 0
ln 2 − x 2 = 3
Alg 3 Functions
33
Answers
(1) 3 , 7
(2) 2 , 3 , ( reject − 5 )
(3) 1 , 34 , ( reject − 3)
(4) 3 , (reject 1)
(
(5) 2 , reject − 1 , 13
(6) 7 , (reject − 3)
(7) 9 , (reject 2)
(8) 8 , 1
4
(9) 1 , 1
32
(10) Ø
)
Alg 3 Functions
34
Algebra 3 Assignment # 8
Calculator Logarithm Problems
(1) Use a calculator to solve each of the following correct to 4 decimal places please.
(2) Let
(a)
5x = 20
(b)
43x + 1 = 91 − x
(c)
log 3 (18) = x
(d)
log 7 ( x ) = 1.432
(e)
ln ( x ) = 1.432
(f)
log ( x )
5 3 = 11
(g)
0.3x > 7
(h) 2
log10 ( 2 ) = p
(a)
log10 ( 6 )
(c)
(e)
and
( ln ( x ) )2 −
5 ln ( x ) − 3 = 0
log10 ( 3) = q . Evaluate each of the following in terms of p and q.
(b)
log10 ( 72 )
log10  5
(d)
log10 ( 90 )
log10 ( 0.5 )
(f)
log10 ( 5 )
3 3

 16 
(3) Simplify the following expression please.
log 4 (125 ) i log 49 ( 32 ) i log 25 ( 7 )
(4) The magnitude of an earthquake is measured using the Richter scale;
M=
 E 
2
log  4.4  ,
3
 10 
Where M is the magnitude of the earthquake, and E is the seismic energy released by the earthquake (in
15
joules). The 1989 San Francisco earthquake released approximately 1.12 x 10 joules. Calculate the
magnitude of the earthquake using the Richter scale. How much energy would be released (in joules) by an
earthquake which measures 8.3 on the Richter scale?
Alg 3 Functions
35
Answers
(1)
(2)
(a) 1.8614
(b) 0.1276
(c) 2.6309
(d) 16.2248
(e) 4.1871
(f) 5.1388
(g) x < −1.6162
(h) 0.6065 , 20.0855
(a) p + q
(b) 3p + 2q
(c) 3 q − 4 p
2
5
(d) 2q + 1
(e) −p
(3) 15
8
16
(4) 7.1 , 7.079 x 10 joules
(f) 1 − p
Alg 3 Functions
36
Algebra 3 Review Worksheet
(1) Solve for x please.
(a) 9
2– x
(b) 8
= 27 2x+1
2x –5
(c) 4 i 8
(i) 9
=
( )
(d) log8 3 4
( 161 )
1–x
4
1
2
(f) log x (16 ) = –
(
)
2
– 2 log8 (x) – 1 = 0
(m) log 2 ( x + 1) + log 2 ( 3x–1) = 5
4
3
(n) log 2 ( x – 3) – log 2 ( x+1) = log 2 ( 8 )
(g) log x (.125 ) = 3
(h) log3 log8 ( x )
(k) 3 ( log8 (x) )
(l) log5 ( 2x + 3) = log5 (1 – x )
= x
(e) log 1 ( x ) = –
= x
log ( 4 )
log 4
(j) 3 9
– 9 3( ) = x
= 16 x+1
2x
log9 ( 4 )
= –1
(o) log 7 ( x + 1) + log 7 ( x ) + log 7 ( 2x + 1) = log 7 ( 30 )
(p) 4
log 4 ( 2 )
+ 4
log 2
( 6)
(2) Use a calculator to solve for x. Express answers correct to 3 decimal places.
x
(a) 3 = 8
3x–2
(b) 2
= 51–x
(c) log3 ( 2 ) = x
(d)
( ln ( x ) )3 −
log 4 ( x )
= 8
4 ln ( x ) = 0
Alg 3 Functions
37
Answers
(1)
(a)
1
8
(i) 4
(b)
19
2
(j) –14
(c) –3
(d)
2
9
(e) 2
(2)
(k) 8 ,
(l) –
1
2
2
3
(m) 3
(f)
1
8
(n) ∅
(g)
1
2
(o) 2
(h) 2
(p) 4
(a) 1.893
(b) 0.812
(c) 0.631
(d) 1 , 7.389 , 0.135
Alg 3 Functions
Algebra 3/Trig
Extra Optional Review –
38
B. Solve for x:
( )
1. 92 − x = 27 2x +1
2. 82 x −5 =
4. log 8 3 4 = x
5. log 1 x = −
4
7. log x (0.125) = 3
10. 3
log 9 4
−9
log 3 4
2
x +1
1
2
8. log 3 (log 8 x) = −1
=x
1− x
 1 
3. 4 ⋅ 82x =  
 16 
6. log x 2 = −
9. 9
log 9 4
1
2
=x
11. 3(log 8 x)2 − 2 log 8 x − 1 = 0
12. log 5 (2x + 3) = log 5 (1 − x)
13. log (1 − x) + log (1 − 2x) = log 3 3
14. log 2 (x − 3) − log 2 (x + 1) = log 2 8
15. log 4 (2x + 3) = log 2 x
16. 4
log 4 2
+ 4log2
6
=8
log 4 x
17. log 7 (x + 1) + log 7 x + log 7 (2x + 1) = log 7 30
C. Isolate x completely.
1. 3 x = 8
2. 23x −2 = 51− x
3. 52x + 3 = 3 x +1
Alg 3 Functions
ANSWERS
1
B. 1. x =
8
39
2. x =
31
11
1
4
4. x =
3. x = -3
1
2
5. x = 2
6. x =
9. x = 4
10. x = -14
1
11. x = ,8
2
12. x = −
14. No solution
15. x = 3
16. x = 4
13. x = −
3
2
7. x =
2
9
8. x = 2
2
3
17. x = 2 (the other 3 are imaginary)
log 8
= 1.893
C1.
log 3
log 20
= .812
2.
log 40
3
125
= −1.759
25
log
3
log
3.