TEST 6 Review 1) - simulation-math.com fileTEST 6 Review 1) 25+3x = 1/16 25+3x = 0.0625 log 2 [2 ......
Transcript of TEST 6 Review 1) - simulation-math.com fileTEST 6 Review 1) 25+3x = 1/16 25+3x = 0.0625 log 2 [2 ......
TEST 6 Review
1)
25+3x
= 1/16
25+3x
= 0.0625
log2[25+3x
] = log2[0.0625] Take base-2 log on both sides.
[5+3x]log2(2) = log10(0.0625)/log10(2) Use Power Rule and change-of-base formula
5+3x = log10(0.0625)/log10(2) Note: log2(2) = 1
5+3x = -4
5+3x - 5 = -4 – 5
3x = -9
x = -3
2) y = 4(4x-2)
3) y = 2x
4) (4/5)x = 16/25
0.8x = 0.64
log0.8[0.8x] = log0.8[0.64] Take base-0.8 log of both sides.
xlog0.8(0.8) = log10(0.64)/log10(0.8) Use Power Rule and change-of-base formula
x = log10(0.64)/log10(0.8) Note: log0.8(0.8) = 1
x = 2
5) 41+2x
= 64
log4[41+2x
] = log4[64] Take base-4 log of both sides.
[1+2x]log4(4) = log10(64)/log10(4) Use Power Rule and change-of-base formula
1+2x = log10(64)/log10(4) Note: log4(4) = 1
1+2x = 3
1+2x - 1= 3 – 1
2x = 2
X = 1
6) f(x) = ax
If a is allowed to be negative, the function could not be defined for
all values of x, such as negative values.)
7) y = (1/2)x
8) 4x = 1/16
4x = 0.015625
log4[4x] = log4[0.015625] Take base-4 log of both sides.
xlog4(4) = log10(0.015625)/log10(4) Use Power Rule and change-of-base formula
x = log10(0.015625)/log10(4) Note: log4(4) = 1
x = -3
3
3 1 1/33
3
1/3
27
9) Write 27 3 in log form
Note: 27 27 27
27 3
27 3
log 3 1/3
10) Log4(7)
When t = 0, f(t) = 8
12) log5(1/25) = -2
5-2 = 1/25 13) 62 = 36 log6(36) = 2
14) Evaluate log8(1/512)
= log10(1/512) / log10(8) = -3
15)
10
5
10
1log
1 25Evaluate log 2
25 log 5
16) Write 43 = 64 in log form log4(64) = 3 17) Write logt(t) - logt(s) + 3 logt(u) as a single logarithm. logt(t) - logt(s) + logt(u)3
3
t t t
3
t t
3
t
log (t) + log (u) - log (s)
log (t u ) - log (s)
t ulog
s
8 9
5 2
1/8 1/9
5 2
1/8 1/9 2
5 5 5
5 5 5
18) Write log as sum and/or difference of logarithms.
log
log log log
1 1log log 2log
8 9
x y
z
x y
z
x y z
x y z
2
b
2
log 49
b
log
log 49
19) Evaluate 2
log property: log
log property: b
Thus, 2 49
x
x
b x
x
2
6n 3
12
6n 3
1/ 62
n 3
2
n 3
2 3
n n n
n n n
n n n
n n
920) Write log as sum and/or difference of logarithms.
9log
9log
1 9log
6
1log 9 log - log
6
1log 9 2log - 3log
6
1 2 3log 9 log log
6 6 6
1 1log 9 log
6 3
x
z
x
z
x
z
x
z
x z
x z
x z
n
1log
2x z
21)Write log log as a single logarithm.
log
m m
m
m n
m n
12
1/ 2
12
1/ 2
12 12 12
112 12 122
1722)Write log as sum and/or difference of logarithms.
17log
log 17 log log
log 17 log log
x
y
x
y
x y
x y
a a a a
3/5 1/ 2 6
a a a a
1/ 2 3/5 6
a a a a
1/ 2 3/5 6
a a a a
1/ 2 3/5 6
a a
1/ 2
a 3/5 6
3 123) Write log log log 6log as a single logarithm.
5 2
log log log log
log log log log
log log log +log
log log
log
x y w z
x y w z
x w y z
x w y z
x w y z
x w
y z
2 5
4
2 5
4 4 4
4 4 4
24)Write log as sum and/or difference of logarithms.7
log log log 7
2log 5log log 7
x y
x y
x y
25) ln 0.986 = -0.0140989243795
10
26) log(0.00314)
log (0.00314) 2.503070351926
6 6
2
6 6
27) Solve log ( 1) log ( 4) 2
Note: 2 written in terms of log base-6 is
2 = log (6 ) log (36)
x x
6 6
6 6 6
6 6
2
2
2
2
2
log ( 1) log ( 4) 2
log ( 1) log ( 4) log (36)
log ( 1) ( 4) log (36)
( 1) ( 4) 36
4 4 36
3 4 36
3 4 36 36 36
3 40 0
3 40 0
( 8)( 5) 0
set x + 8 = 0 set x -
x x
x x
x x
x x
x x x
x x
x x
x x
x x
x x
5 = 0
x = -8 x = 5
6 6
6 6
6 6
6 6
6
Check answers:
For x = -8
log ( 1) log ( 4) 2
log ( 8 1) log ( 8 4) 2
log ( 9) log ( 4) 2
Since log of a negative number is undefined,
-8 is an extraneous solution.
For x = 5
log ( 1) log ( 4) 2
log (5 1)
x x
x x
6
6 6
6
6
log (5 4) 2
log (4) log (9) 2
log (4 9) 2
log (36) 2
2 2
Solution set is {5}
5 5
3
5 5
5 5 5
5 5 5
5 5
28) Solve log ( 1) 3 log (6 4)
Note: 3 written in terms of log base-5 is
3 = log (5 ) log (125)
log ( 1) log (125) log (6 4)
log ( 1) log (125) log (6 4)
1log log (6 4)
125
1
1
x x
x x
x x
xx
x
6 425
1(125) 6 4 125
125
1 750 500
1 750 500
0 750 500
0 749 501
0 501 749 501 501
501 749
501 749
749 749
501
7
1 1
49
1
x
xx
x x
x x
x
x
x
x
x
x
x
x
x
5 5
5 5
5 5
Check Answer:
501
749
log ( 1) 3 log (6 4)
501 501log 1 3 log 6 4
749 749
501log negative number 3 log 6 4
749
501 is an extraneous solution.
749
x
x x
x
229) ln(4.37 10 ) ln(0.0437) 3.130407176
1030) log 2.18 log 2.18 0.3384564936
31) 193-x = 26
log19[193-x
] = log19[26] Take base-19 log of both sides.
[3-x]log19(19) = log10(26)/log10(19) Use Power Rule and change-of-base formula
3-x = log10(26)/log10(19) Note: log19(19) = 1
3-x = 1.10652540639298
3-x-3= 1.10652540639298 -3
-x = -1.89347459360702
X = 1.89347459360702
32) 6x+1 = 29
log6[6x+1
] = log6[29] Take base-6 log of both sides.
[x+1]log6(6) = log10(29)/log10(6) Use Power Rule and change-of-base formula
x+1 = log10(29)/log10(6) Note: log6(6) = 1
x+1 = 1.87932358545715
x+1-1 = 1.87932358545715 -1
x = 0.87932358545715;
33) 20x = 54
log20[20x] = log20[54] Take base-20 log of both sides.
xlog20(20) = log10(54)/log10(20) Use Power Rule and change-of-base formula
x = log10(54)/log10(20) Note: log20(20) = 1
x = 1.33155558718601
34)
log₄(x²) = log₄(5x + 14)
(x²) = (5x+14)
x² = 5x + 14
x² - (5x + 14) = 5x + 14 - (5x + 14)
x²-5x - 14 = 0
(x-7)(x+2) = 0
Set x-7 = 0 set x+2 = 0
X = 7 x = -2
Check answers:
log₄(x²) = log₄(5x + 14)
log₄((7)²) = log₄(5(7) + 14)
log₄(49) = log₄(49)
log₄((-2)²) = log₄(5(-2) + 14)
log₄(4) = log₄(4)
Solution set is {7, -2}
35)
log3(x) = 5
Note: 5 can be written as 5 = log3(35) = log3(243)
Rewrite log3(x) = 5 as follows:
log3(x) = log3(243)
x = 243
Solution set is {243}
Check answer:
log3(x) = 5
log3(243) = 5
5=5