Expert: Sherman D. - 3/15/2005

Question
determine whether each x-value is a solution (or an approximate solution) of the equation.

1.)4^(2x-7)=64
 a. x=5
 b. x=2
2.)2^(3x+1)=32
 a. x=-1
 b. x=2
3.)3e^(x+2)=75
 a. x=-2+e^25
 b. x=-2+ln 25
 c. x approximately = 1.219
5.) log[4](3x)=3
 a. x approximately = 20.356
 b. x=-4
 c. x= 64/3
6.) ln(x-1)=3.8
 a. x= 1+e^3.8
 b. x approximately= 45.701
 c. x=1+ ln 3.8

solve for x

7.  4^x=16
8.  3^x =243
9.  5^x =625
10. 3^x=729
11. 7^x=1/49
12. 8^x=4
13. (1/2)^x=32
14. (1/4)^x=64
15. (3/4)^x= 27/64
16. (2/3)^x=4/9
17. ln x - ln 2=0
18. ln x - ln 5 = 0
19. e^x=2
20. e^x=4
21. ln x= -1
22. ln x = -7
23. log[4] x=3
24. log[5]x=-3
25. log[10]x=-1
26. log[10]x-2=0

approximate point of intersection of the graphhs of f and g. Then solver the equation f(x)=g(x) algebraically.

27. f(x)=2^2
   g(x)=8
28. f(x)=27^x
   g(x)=9
29. f(x)=log[3]x
   g(x)=2
30. f(x)=ln(x-4)
   g(x)=0

Answer
1.)
4^(2x - 7) = 64
4^(2x - 7) = 4^3
therefore
2x - 7 = 3
2x = 10
x = 5
ANS :  
a. x = 5

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2.)
2^(3x + 1) = 32
2^(3x + 1) = 2^5
3x + 1 = 5
3x = 4
x = (4/3)
You may have something written wrong here, but thats what i got.
a. x=-1
b. x=2

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3.)
3e^(x + 2) = 75
e^(x + 2) = 25
log(e)25 = x + 2
ln(25) = x + 2
x = ln(25) - 2
ANS :
b. x = -2 + ln(25)

Don't see a #4 in case you left it out.

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5.)
log[4](3x) = 3
4^3 = 3x
3x = 64
x = (64/3)
ANS :
c. x = 64/3

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6.)
ln(x - 1) = 3.8
log(e)(x - 1) = 3.8
e^3.8 = x - 1
x = e^(3.8) + 1
ANS :  
a. x = 1 + e^3.8
b. x approximately = 45.701

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7.)
4^x = 16
4^x = 4^2
x = 2

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8.)
3^x = 243
3^x = 3^5
x = 5

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9.)
5^x = 625
5^x = 5^4
x = 4

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10.)
3^x = 729
3^x = 3^6
x = 6

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11.)
7^x = (1/49)
7^x = 49^-1
7^x = (7^2)^-1
7^x = 7^(2 * -1)
7^x = 7^(-2)
x = -2

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12.)
8^x = 4
(2^3)^x = (2^2)
2^(3x) = 2^2
3x = 2
x = (2/3)

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13.)
(1/2)^x = 32
(2^-1)^x = 32
2^(-x) = 32
2^(-x) = 2^5
-x = 5
x = -5

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14.)
(1/4)^x = 64
(4^-1)^x = 64
4^(-x) = 64
4^(-x) = 4^3
-x = 3
x = -3

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15.)
(3/4)^x = 27/64
same as saying
(3^x)/(4^x) = 27/64
3^x = 27
3^x = 3^3
x = 3

4^x = 64
4^x = 4^3
x = 3

so

x = 3

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16.)
(2/3)^x = 4/9
Same reason about
x = 2

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17.)
ln(x) - ln(2) = 0
ln(x) = ln(2)
x = 2

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18.)
ln x - ln 5 = 0
ln(x) = ln(5)
x = 5

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19.)
e^x = 2
log(e)2 = x
x = ln(2)

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20.)
e^x = 4
log(e)4 = x
x = ln(4)

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21.)
ln(x) = -1
log(e)x = -1
(log(x))/(log(e)) = -1
log(x) = -log(e)
log(x) = log(e^(-1))
x = e^(-1)

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22.)
ln x = -7
log(e)x = -7
(log(x))/(log(e)) = -7
log(x) = -7(log(e))
log(x) = log(e^(-7))
x = e^(-7)

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23.)
log[4] x = 3
4^3 = x
x = 64

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24.)
log[5] x = -3
5^(-3) = x
x = (1/125)

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25.)
log[10]x = -1
10^-1 = x
x = (1/10)

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26.)
log[10]x - 2 = 0
log(10)x = 2
10^2 = x
x = 100
unless you meant
log(10)(x - 2) = 0
10^0 = x - 2
1 = x - 2
x = 3

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27.)
f(x) = 2^2
g(x) = 8

Their Parallel, so therefore they will never meet.

if you mean

f(x) = 2^x
g(x) = 8

then the answer is (3,8)

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28.)
f(x) = 27^x
g(x) = 9

27^x = 9
(3^3)^x = (3^2)
3^(3x) = 3^2
3x = 2
x = (2/3)

The point would be ((2/3),9)

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29.)
f(x) = log[3]x
g(x) = 2

log(3)x = 2
3^2 = x
x = 9

The point would be (9,2)

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30.)
f(x) = ln(x - 4)
g(x) = 0

ln(x - 4) = 0
log(e)(x - 4) = 0
e^0 = x - 4
1 = x - 4
x = 5

Point (5,0)