Download answers - Exponential and Logarithmic functions

Self-assessment B – answers - Exponential and Logarithmic functions
Name: _____________________________________
1. What is the inverse of f (x )  5
x
1. f 1 ( x)  log 5 x ______
f ( x)  5 x
Step 1. y  5 x
Step 2. x  5 y
Step 3. y  log 5 x
Step 4. f 1 ( x )  log 5 x
2. What is the domain of y=Log5x
2. _positive numbers, that is,
(0,  ) _________
3. Give the number e to the nearest 6 decimal places.
3. _2.718282_________
x
4. If 8  32 , find x
8 x  32  2 3x  2 5  3x  5  x 
4. 1.66666666…____
5
 1.66666....
3
or
x  log 8 32 
5. If 2
x3
log 32 1.505149978

 1.666666....
log 8 0.903089987
 10 , find x (six decimal places.)
2 x  3  10  x  3  log 2 10  x  3 
5. x= 6.32193________
ln 10
ln 2
 x  3  3.32193  6.32193
6. If lnx = -1.6531472, then x=__
6. 0.1914464393___
lnx  - 1.6531472  x  invln(-1.6 531472)  e
-1.6531472
 x  0.1914464393
7. Find Log1.78 4.91 (5 decimal places)
log 4.91 ln 4.91
log 1.78 4.91 

 2.75969
log 1.78
ln 1.78
7. 2.75969_______
9. If Log2 (x+1) + Log2 5 = 4, find x
log 2 ( x  1)  log 2 5  4  log 2 5( x  1)  4  5( x  1)  2 4  5x  5  16
11
 5x  11  x 
5
-1-
a
10. If log x=a and log y=b, give log (x b y ) in terms of a and b.
log( x a b y )  a log x 
log y
b
 a  a   a2  1
b
b
a
10. __ log (x b y ) = a 2  1 ____
11. Find x without using calculators if 4
x3
x 2
8
4 x  3  8 x  2  2 2( x  3)  23( x  2)  2( x  3)  3( x  2)  2x  6  3x  6
  x  12  x  12
11. __x = 12__________
12-15. If Log5 2=0.430 and Log53=0.682, find (Do not use calculators)
12. Log5 32
12. 2.150
log 5 32  log 5 2 5  5 log 5 2  5(0.430)  2.150
13) Log2 5
log 5 5
1
100
log 2 5 


log 5 2 0.430 43
14) Log5 18
13.
100
43
14. 1.794
log 5 18  log 5 2  32  log 5 2  2 log 5 3  0.430  2(0.682)
 0.430  1.364  1.794
15) Log5
log 5
4
3
15. 0.178
4
 log 5 4  log 5 3  log 5 2 2  log 5 3  2 log 5 2  log 5 3
3
 2(0.430)  0.682  0.860  0.682  0.178
16. If 2Log4(x-5) =1 then find x.
2 log 4 (x  5)  1  log 4 (x  5) 
1
 x  5  41 / 2  x  5  4  5  2  7
2
16. ___x = 7_____________
(2. 38)5.3
17. Use Ln to compute
3
3.8
x
( 2.38) 5.3
3 3.8
 log x  5.3 log( 2.38) 
log( 3.8)
 log x  1.802596674
3
 x  inv log(1.802596674)  101.802596674  63.4741178
17. _63.4741178____________________
-2-
18. If Log (x-3) + log(x-4) =Log 20, find x.
Log(x - 3)(x -4) = log 20
(x – 3)(x – 4) = 20
x 2  7x  12  20  x 2  7x  8  0  x  8x  1  0  x  8 or x  1
x = -1 is not possible because the domain of the log function is the positive numbers.
Therefore x = 8
18. _x = 8_______________
19. If $30,000 is invested at 7.5% interest compounded monthly for 20 years, what is
the balance after 20 years?
r

Use the formula B  P 1  
 n
r

B  P 1  
 n
nt
nt
answer: $133,824.51_____
12 ( 20 )
0.075 

 B  30000 1 

12 

 133824.51
20. In how many years will the investment of problem 19 will give a balance of
$150,000? You can use a calculator.
r

B  P 1  
 n
nt
12 t
0.075 

 150000  30000 1 
 5  (1.00625)12 t

12 

log 5
 12t  log 1.00625 5  12t 
 12t  258.3139493
log 1.00625
t
258.3139493
 21.526 years
12
21. Find x if log 1 (x  3)  2
3
1
log 1 ( x  3)  2  x  3   
 3
2
 x 3  32  x 3  9  x  6
3
21. __x = 6_______
22. Find x if log5x-log3=log(2x+3)
5x
5x
log
 log( 2x  3) 
 2x  3  5x  6x  9  x  9
3
3
Answer: no solution because domain of the log function is the positive numbers
23. Find x if Log x8 = 3.
x3  8  x  2
23. _x = 2______
-3-
24. Write with only one Ln:
5Lnx + 2Lny -3Lnz - 3Lnw
5Lnx  2Lny - 3Lnz - 3Lnw  Ln
x5y 2
z 3w 3
24. Ln
x5y 2
z 3w 3
_______
25. Find x if Log9 x =-2
log 9 x  2  x  9  2  x 
1
9
2

1
81
25. ___x =
-4-
1
____
81