Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Any suggestion/request : write to rk01970@gmail.com 
Any comment: on www.facebook.com/eblackboard 
Presentation file : www...
www.slideshare.net/eblackboard
1. 
Express the following relations in the logarithmic forms: 
Ans. 
HOME 
Solution: 
3 81 
log 81 4 
3 
4 
 
 
Logarith...
1. 
Express the following relations in the logarithmic forms: 
Ans. 
HOME 
Solution: 
5 1 
log 1 0 
5 
0 
 
 
Logarithmi...
1. 
Express the following relations in the logarithmic forms: 
Ans. 
HOME 
Solution: 
1 
5 
32 2 
1 
5 
 
32 2 
log 2 
32...
1. 
Express the following relations in the logarithmic forms: 
Ans. 
HOME 
Solution: 
3 
1 
1 
125 
log 
125 
5 
5 
3 
 ...
1. 
Express the following relations in the logarithmic forms: 
Logarithmic form : 
Ans. 
HOME 
Solution: 
a c 
c b 
a 
b 
...
2. 
Express the following logarithmic forms in the exponential forms: 
log 64 6 
Ans. 
HOME 
Solution: 
6 
2 
2  
64 
 
...
2. 
Express the following logarithmic forms in the exponential forms: 
exponential form : 
Ans. 
HOME 
Solution: 
1 
1 
81...
2. 
Express the following logarithmic forms in the exponential forms: 
log 1 0 
Ans. 
HOME 
Solution: 
1 
0  
 
a 
a 
ex...
2. 
Express the following logarithmic forms in the exponential forms: 
3 
exponential form : 
Ans. 
HOME 
Solution: 
1 
1 ...
2. 
Express the following logarithmic forms in the exponential forms: 
P R 
log  
exponential form : 
Ans. 
HOME 
Solutio...
3. 
Find the value of x : 
log 128 
Ans. 
HOME 
Solution: 
log 128 
  
2 128 
, 2  
2 
, 7 
7 
2 
2 
 
 
 
or 
or x ...
3. 
Find the value of x : 
log 81 4 
log 81 4 
Ans. 
HOME 
Solution: 
81 
x 
  
4 4 
4 
or x 
,  
3 
,  
3 
 
 
or x...
3. 
Find the value of x : 
Ans. 
HOME 
Solution: 
log   
3 
x 
log 3 
or x 
1 
 
125 
 5 
 
, 
1 
5 
, 
3 
3 
5 
5 
...
3. 
Find the value of x : 
log 243 10 
log 243 10 
Ans. 
HOME 
Solution: 
10 5 
3 
x 
  
or x 
,  
3 
or x 
, 3 
1 
2 
...
3. 
Ans. 
HOME 
Find the value of x : 
Solution: 
log 49 
7 
 
log 49 
7 
  
  
 
7 49 
1 
 
2 2 
, 7 7 
2 
1 
, 
o...
3. 
Find the value of x : 
log 7 5 2 
10 
  
x 
log 7 5 2 
2 
10 
   
10 7 5 
or x 
, 100  7  
5 
or , 100  5  
7...
3. 
Find the value of x : 
log 0.0001 
log 0.0001 
10 
log 10 
4 
 
 
x 
x 
(vii) x 
or x 
, 4log 10 
Ans. 
HOME 
Soluti...
3. 
5 
 
  
Ans. 
HOME 
Find the value of x : 
Solution: 
log 0.25 4 
log 0.25 4 
  
 
1 
2 
 
 
5 
 
 
5 
10 
x...
4. 
Find the value of logarithms of : 
(i) to the base of 
625 5 
log 625 
5 
log 5 
, 4log 5 
Ans. 
HOME 
Solution: 
, 4 ...
4. 
Find the value of logarithms of : 
343 7 
log 343 
7 
log 7 
, log 7 
, log 7 
6 
, 6log 7 log 7 1 
Ans. 
HOME 
Soluti...
4. 
Find the value of logarithms of : 
to the base of 
  
Ans. 
HOME 
Solution: 
0.1 9 3 
  
log 0.1 
1 
let x let 
9 ...
4. 
Find the value of logarithms of : 
to the base of 
1728 2 3 
log 1728 
2 3 
  
log 2 3 
6 3 
let log 2  3 
 
x 
6 ...
4. 
Find the value of logarithms of : 
to the base of 
2401 7 
log 2401 
7 
3 
log 7 
let log 7 
x 
x 
3 4 
, 7 7 
1 
 
3...
4. 
Find the value of logarithms of : 
to the base of 
2 4 
Ans. 
HOME 
Solution: 
log 2 
let x 
4 
8 
4 
 
 
, 4  
2 
...
4. 
Find the value of logarithms of : 
Ans. HOME 
Solution: 
to the base of 
81 9 
log 81 
9 
let log 81 
x 
9 
 3 
 
x ...
4. 
Find the value of logarithms of : 
to the base of 
5 0.008 
log 5 
0.008 
let x 
x 
, 0.008 5 
1 
 
  
1 
1 
1 
An...
5. 
Find the base when: 
is the logarithamof 
3 343 
let base x 
  
log 343 3 
3 
or x 
,  
343 
3 3 
Ans. 
HOME 
Solut...
5. 
Find the base when: 
is the logarithamof 
4 144 
let base  
x 
  
log 144 4 
4 
or x 
,  
144 
4 4 2 
or x 
, 2 3 ...
5. 
Find the base when: 
is the logarithamof 
let base  
x 
   
1 
1 
Ans. 
HOME 
Solution: 
  
1 
3 
1 
1 
 
 
 ...
5. 
Find the base when: 
let base  
x 
   
Ans. 
HOME 
Solution: 
 
 
1 
or , 
x a 
or x a 
a 
or x 
a 
a 
is the lo...
6. 
Find the simplest value of : 
log 5 log 27 
log 27 
10 
log 3 
3log 3 
3 25 
log 5 
10 
log 5 
10 
log 5 
 
 
Ans. 
...
6. 
Find the simplest value of : 
log 27 if log 3 
8 2 
log 27 
10 
log 8 
10 
log 3 
log 2 
3log 3 
10 
3log 2 
or, 
or, ...
6. 
log 
log 
log 
1 
2 
1 
x 
x 
log 
Ans. 
HOME 
Find the simplest value of : 
(iii) 
x x a 
log if log 
Solution: 
2 2 ...
7. 
Prove that log(1 23)  log1 log2  log3 
proved 
HOME 
Solution: 
L H S 
. . . 
log(1  2  
3) 
log(6) 
log(1  2 ...
8. 
Express M in term of N 
M N 
log 3log 1 
2 
1 
  
M N 
  
M N 
or, log  log  
1 
 
 
or, log log 3 log 1 
solv...
8. 
Express M in term of N 
(ii)   
N M 
log 3 2log 
10 10 
N   
M 
N M 
or, log  log  
3 
N M 
or, log (  )  
3 
...
9. 
Prove that : 
log 
log 
x 
a  
x 
let y a 
x 
y a 
  
log log 
. ..(i) 
log 
x 
x a 
a a 
log log 
a a 
x a 
 
 ...
9. 
Prove that : 
a 
2log 2 
2log 
x  
a 
let y x 
a 
y x 
  
log log 
. ..(i) 
2log 
a 
a x 
x x 
2log log 
x x 
a x 
...
9. 
y x 
log log 
log 
y 
x y 
(iii)  
z x 
  
log log 
. ..(i) 
log 
y 
a a 
a 
y x 
log log 
a a 
log log 
 
 
z y ...
9. 
m n n m 
(iv)    
log log log log 
a b a b 
n 
b 
m 
b 
L H S 
m  
n 
a b 
m 
a 
n 
a 
 
 
n m R H S 
proved 
HO...
9. 
log 3  log 2  
1 
1 
2 3 
. . . 
log 3 log 2 
proved 
HOME 
Prove that : 
Solution: 
2 3 
or, 1 . . . 
] 
1 
log 
[ ...
9. 
b c a 
log  log  log  
1 
a b c 
. . . 
b c a 
log  log  
log 
log 
c 
proved 
HOME 
Prove that : 
Solution: 
log...
9. 
Prove that : 
    
2 2 
log log log( ) log 
. . . 
x y 
log log 
x 
 
 
x y x y a b a b a b 
or, log  log log ...
10. 
Find the value of : 
(i)  
log 729 9 (27) 
4 3 
log 729 9 (27) 
4 
4 
 
or, log 3 3 (3 ) 
4 6 3  2  
4 
3 
or, lo...
10. 
Find the value of : 
log log log 81 
3 2 3 
log log log 81 
3 2 3 
or, log log log 3 [  
3  
81] 
or, log log 8log ...
10. 
Find the value of : 
log 16  
log 9 
2 3 
log 16 log 9 
2 3 
or, log 2  log 3 [  
2  16 3  
9] 
or, 8log 2 4log ...
10. 
Find the value of : 
b c d a 
(iv)    
log log log log 
a b c d 
b c d a 
log  log  log  
log 
a b c d 
log 
b ...
10. 
Find the value of : 
(v)   
log 27 log 8 log 1000 
log1.2 
log 27  log 8  
log 1000 
3 3 3 
log 3  log 2  
log ...
10. 
Find the value of : 
(vi)     
log 4 log 5 log 6 log 7 log 3 
3 4 5 6 7 
log 4  log 5  log 6  log 7  
log 3 
...
10. 
Find the value of : 
log 6 6 6... 
x 
let  6 6 6...  
. ..(i) 
2 
x 
or, 6 6 6 6... 
  
x x 
or,  
6 from (i) 
o...
11. 
25 
    
25 
   
16 12 7 
 
5 
25 
5 
16 
16 
 
16 
2 
2 
25 
16 
. . . 
 
24 
 
 
  
64 
 
 
 
 
...
11. 
   
  
7 2 3 
25 
 
25 
 
 
  
 
  
7 2 
5 
5 
4 
 
6 2 
2 3 
10 
 
10 
 
  
5 2 
 
7 7 
5 2 
7 ...
11. 
   
  
7 5 3 
25 
 
25 
 
 
  
10 
5 
16 
 
16 
2 
2 
5 
81 
81 
81 
25 
25 
16 
. . . 
16 
 
 
 
 
1...
11. 
32 
   
32 
 
5 
2 
2 
2 
 
  
 
   
 
5 
3 
75 
5 
5 
 
  
3 5 
3 5 
3 5 
5 
75 
. . . 
75 
 
  
...
11. 
(v) x y z 
y z z x x y 
log  log log  log log  
log 
   
y z z x x y 
log  log log  log log  
log 
let k  x...
11. 
Prove that: 
1 
   
xy yz zx 
1 
1 
. . . 
1 
xyz xyz xyz 
xy yz zx 
   
xy yz zx 
or, log ( ) 
xyz 
xyz 
or, l...
11. 
a b c 
   
3 3 3 
b c a 
. . . 
a b c 
log log log 
3 3 3 
c 
b 
   
1 
log 
log 
b 
proved 
HOME 
Prove that: ...
11. 
. . . 
2 3 
a a a a 
log  log  log  ...  
log 
n 
a a a n a 
or, log  2log  3log  ...  
log 
or, 1  2  3  ...
11. 
x y z x y z 
(ix)      
log log log log log log 
a b c b c a 
proved 
HOME 
Prove that: 
Solution: 
z 
c 
z 
a 
...
11. 
Prove that: 
x y z 
log  log  log   
1 
1 1 1 
y z x 
. . . 
x y z 
log  log  
log 
1 1 1 
log 
y 
  
log 
y ...
11. 
Prove that: 
x y z 
   
2 2 2 
x y z 
. . . 
x y z 
log log log 
2 2 2 
x y z 
   
1 
1 
1 
1 
  
proved 
HOM...
12 (a) 
log 25 
8 
log 25 
log 8 
log 5 
2 
log 2 
3 
2log 5 
3log 2 
10 
2 
2log 
 
3log 2 
 
2 log10 log 2 
3log 2 
2 ...
12 (b) 
log 3 , log 5 , log 8. 30 30 30 If  a and  b find the valueof 
 
30  
log 8 
log 2 
3log 2 
 
30 
30 
  
 ...
13. (i) 
1 
1 
 
 
If a 2  b  
2  7 ab , show that, log a  
b  log a  log b 
a  b  
ab 
2 2 
a b ab ab ab 
o...
13. (ii) 
1 
y 
x 
log log  , 23 
2 
 
log     
  
  
  
  
1 
  
2 2 
25 
 
2 
or, 
2 2 
x y xy xy 
or,...
13. (iii) 
 
 
If a b a b showthat x x x x x , log log 3 5 5 3   
x x x x 
3  5 5  
3 
   
x 
x 
5 
3 
5 
3 
 
...
13. (iv) 
4 4 2 2 
a  b  
14 
a b 
4 4 2 2 2 2 2 2 
a b a b a b a b 
or,   2  14  
2 
  
    
2 2 2 2 2 
a b a...
14. (a) 
y 
y 
x 
x 
log log log 
  
  
  
x k y z 
   
z 
z 
log . ... .(i) 
y k z x 
log   
. ... .(ii) 
z k ...
14. (b) (i) 
x 
y 
x 
log log log 
y 
z 
z 
    
    
    
x k b c a x ak b c 
log log . ... .(i) 
     ...
14. (b) (ii) 
y 
x 
y 
x 
log log log 
 
z 
z 
         2 2 
 
          
          
x k b ...
14. (b) (iii) 
y 
x 
y 
x 
log log log 
z 
z 
    
    
    
x k ry qz p x pk ry qz 
      
log log . .....
15. 
log 45 
2 
  
log(3 5) 
2 
  
log 3 log 5 
  
2log 3 log 5 
10 
 
 
   
2log 3 log10 log 2 
    
2 0.4...
15. 
log108 
2 3 
  
log(2 3 ) 
2 3 
  
log 2 log 3 
  
2log 2 3log 3 
    
2 0.3010 3 0.4771 
  
0.6020 1.431...
15. 
log 84 
2 
   
log(2 3 7) 
2 
   
log 2 log 3 log 7 
   
2log 2 log 3 log 7 
    
2 0.3010 0.4771 0.8451...
15. 
log 294 
   
log(2 3 7 ) 
   
log 2 log 3 log 7 
   
log 2 log 3 2log 7 
    
0.3010 0.4771 2 0.8451 
 ...
15. 
log 21.6 
216 
10 
log 
 
 
  
log 216 log10 
 3 3 
 
   
log 2 3 log10 
3 3 
   
log 2 log 3 log10 
  ...
16 (i) 
given 
If three positive real numbers a b and c are inG P 
, . ., 
solved 
showthat a b and c are in A P 
a, b and...
16 (ii) 
log 
log 
x k 
log 
  
log 
log 
log 
   
log . ... .(i) 
y k 
log  
2 . ... .(ii) 
z k 
log  
3 . ... .(i...
17. 
The f irst and the last terms of aG P are a and k respectively 
If the number of termbe n provethat 
k a 
log log 
gi...
18. 
If the p q and r terms of aG P are a b and c respectively showthat th th th 
given 
, . . , , 
q  r a  r  p b  p ...
since a, b and c are in G.P. 
- - - (ii) 
- - - (iii) 
- - - (i) 
b 
q p 
 
 
 
  
R 
1 
1 1 
R 
 
1 
q 
1 
1 
1 1 ...
from (i), (ii) and (iii) : 
1 1 
b q p c 
r q 
   
c 
 
  
 
  
q  
p 
q p 
b 
r q 
c 
b 
 
b 
 
 
 
 
r ...
r p b r q a q p c 
(  ) log  (  ) log  (  
) log 
r p b q r a p q c 
(  ) log   (  ) log  (  
) log 
q r a p q ...
19. (i) 
If x  bc y  ca and z  
ab showthat a b c 
1 
x y 1 
z 
1 
x y z 
1 
1 
bc ca ab 
a b c 
bc a ca b ab c 
a a b ...
19. (ii) 
given 
If x  log ( bc ), y  log ( ca ) and z  
log ab , 
showthat a b c 
( ) 2 
x bc 
log ( ) 
a 
 
y ca 
lo...
20. 
given 
a a a 
. . . 
1 
    
log log 2 log 3 1 
a a a 
2 3 4 
a 
    
a 
L H S 
log 
log 
log 2 
  
a a 
lo...
21. 
log  , log  , prove log  
p q p  
given 
p 
q 
1 
p q 
. . . 
1 
x 
log 
log log 
x x 
1 
ab 
proved 
x  
a 
x b...
 
 
If log x y a, and log , log log 2 3    
22. 
Solution: 
given: 
x 
log( x 2 y 3 
) a 
- - - (i) b 
x 
 
 
log -...
23. 
e 
y 
2 
 2 
 
solved 
y y 
e  
e 
1 
If x  y  
y e 
y y 
e e 
 
 
y y 
y y 
e e 
 
 
y y 
e e 
e e 
y  y y...
24. 
log 3 10 Showthat the valueof lies between and 
Solution: 
HOME 
2 
. 
5 
1 
2 
9 10 
log 9 log 10 
10 10 
- - - (i) ...
25. 
Solution: 
HOME 
log 10 log (32 ) 5, . 6 If b and b find the value of a a a   
a  
log 10 (given) 
10 
a b 
log (3...
26. 
Solution: 
HOME 
found
27. (i) Solve 
Solution: 
HOME 
4 
solved 
x 
x x 
2 
10 
log  log  
10 10 log 
2 
x 
 
x 
 
 
10 1 
x 
x 
x 
x 
x 
x...
27. (ii) Solve 
Solution: 
log log log 1 2 2 2 x  
 x 
log log log 1 
2 2 2 
  
log log log log 2 
2 2 2 2 
log log 2 ...
27. (iii) Solve 
Solution: 
HOME 
log log log 11 8 4 2 x  x  x  
log  log  log  
11 
8 4 2 
1 
    
x x x 
1 
 ...
28. 
If a b and c are three consecutive positive integers showthat 
log(1 ) 2log 
Solution: 
HOME 
, , 
  
a b and c are...
Upcoming SlideShare
Loading in …5
×

Logarithm

2,364 views

Published on

Solution of Logarithm,
Book - Business Mathematics and Statistics (S.N.Dey)

Published in: Education
  • Be the first to comment

Logarithm

  1. 1. Any suggestion/request : write to rk01970@gmail.com Any comment: on www.facebook.com/eblackboard Presentation file : www.slideshare.net/eblackboard Contact No. : 8100803074
  2. 2. www.slideshare.net/eblackboard
  3. 3. 1. Express the following relations in the logarithmic forms: Ans. HOME Solution: 3 81 log 81 4 3 4   Logarithmic form : (i)
  4. 4. 1. Express the following relations in the logarithmic forms: Ans. HOME Solution: 5 1 log 1 0 5 0   Logarithmic form : (ii)
  5. 5. 1. Express the following relations in the logarithmic forms: Ans. HOME Solution: 1 5 32 2 1 5  32 2 log 2 32 5   Logarithmic form : (iii)
  6. 6. 1. Express the following relations in the logarithmic forms: Ans. HOME Solution: 3 1 1 125 log 125 5 5 3     Logarithmic form : (iv)
  7. 7. 1. Express the following relations in the logarithmic forms: Logarithmic form : Ans. HOME Solution: a c c b a b   log (v)
  8. 8. 2. Express the following logarithmic forms in the exponential forms: log 64 6 Ans. HOME Solution: 6 2 2  64  exponential form : (i)
  9. 9. 2. Express the following logarithmic forms in the exponential forms: exponential form : Ans. HOME Solution: 1 1 81 3 4 81 log 4 3     (ii)
  10. 10. 2. Express the following logarithmic forms in the exponential forms: log 1 0 Ans. HOME Solution: 1 0   a a exponential form : (iii)
  11. 11. 2. Express the following logarithmic forms in the exponential forms: 3 exponential form : Ans. HOME Solution: 1 1  125 25 2 125 log 25 3 2          (iv)
  12. 12. 2. Express the following logarithmic forms in the exponential forms: P R log  exponential form : Ans. HOME Solution: R Q Q  P (v)
  13. 13. 3. Find the value of x : log 128 Ans. HOME Solution: log 128   2 128 , 2  2 , 7 7 2 2    or or x x x x x  (i)
  14. 14. 3. Find the value of x : log 81 4 log 81 4 Ans. HOME Solution: 81 x   4 4 4 or x ,  3 ,  3   or x x x  (ii)
  15. 15. 3. Find the value of x : Ans. HOME Solution: log   3 x log 3 or x 1  125  5  , 1 5 , 3 3 5 5     or x x x  (iii)
  16. 16. 3. Find the value of x : log 243 10 log 243 10 Ans. HOME Solution: 10 5 3 x   or x ,  3 or x , 3 1 2 10  , 3 1 5     or x x x  (iv)
  17. 17. 3. Ans. HOME Find the value of x : Solution: log 49 7  log 49 7      7 49 1  2 2 , 7 7 2 1 , or or x 2    ,  2  2  4 or x x x x x  (v)
  18. 18. 3. Find the value of x : log 7 5 2 10   x log 7 5 2 2 10    10 7 5 or x , 100  7  5 or , 100  5  7 x or x Ans. HOME Solution:     , 105 7 105 , or x 7 or , 15 x   ,  15    or x x x  (vi)
  19. 19. 3. Find the value of x : log 0.0001 log 0.0001 10 log 10 4   x x (vii) x or x , 4log 10 Ans. HOME Solution:       , 4  log 10 1 10 10 10 10         or x
  20. 20. 3. 5    Ans. HOME Find the value of x : Solution: log 0.25 4 log 0.25 4    1 2   5   5 10 x or x or x , 10 , 10 25 100 , 0.25 2 1 2 2 4 2 4                           or x x x  (viii)
  21. 21. 4. Find the value of logarithms of : (i) to the base of 625 5 log 625 5 log 5 , 4log 5 Ans. HOME Solution: , 4  log 5 1 5 5 4 5     or or
  22. 22. 4. Find the value of logarithms of : 343 7 log 343 7 log 7 , log 7 , log 7 6 , 6log 7 log 7 1 Ans. HOME Solution:         , 6 7 7 7 2 3 7 3 7 or or or or to the base of      (ii)
  23. 23. 4. Find the value of logarithms of : to the base of   Ans. HOME Solution: 0.1 9 3   log 0.1 1 let x let 9 3   1   2 2 4 5 1 1 2  , 3 3 3 1  2 2 2  , 3 3 5       , 3 3 2 5 5 ,   or or or or or x , 2 2 2 9 , 9 3 9 log 1 9 0.1 9 log 2 2 . 9 3 . 9 3 .                                 or x x x x x   (iii)
  24. 24. 4. Find the value of logarithms of : to the base of 1728 2 3 log 1728 2 3   log 2 3 6 3 let log 2  3  x 6 3 x , 2 3 2 3     x , 2 3 2 3 x , 2 3  2  3 Ans. HOME Solution:             6 6   ,    6 2 3 2 3  , 6 3 2 6 2 3 6 3 2 3      or or or or or x x  (iv)
  25. 25. 4. Find the value of logarithms of : to the base of 2401 7 log 2401 7 3 log 7 let log 7 x x 3 4 , 7 7 1  3 4      , 7 7 Ans. HOME Solution:   4 1 , or or or x 3 , 12 4 7 4 7 3 3 3         or x x  (v)
  26. 26. 4. Find the value of logarithms of : to the base of 2 4 Ans. HOME Solution: log 2 let x 4 8 4   , 4  2   4 8 8   2 8 , 2  2 , 2  2  or or or or x , 2   8 8 2 , log 2 2 8 -8      or x x x x  (vi)
  27. 27. 4. Find the value of logarithms of : Ans. HOME Solution: to the base of 81 9 log 81 9 let log 81 x 9  3  x , 9 81 1  3 2      , 9 9 1 3 2 , 9 9 2 1 , or or or or x 3   , 2 3 6 3 3 3         or x x x  (vii)
  28. 28. 4. Find the value of logarithms of : to the base of 5 0.008 log 5 0.008 let x x , 0.008 5 1    1 1 1 Ans. HOME Solution:     8 1 1         , 5 5 , 5 5 1 1 1 1 6 or or or or or or or x or x , 6 1   2 3 , - 2 , -3 5 5 , 5 125 , 5 1000 , log 5 2 -3 2 -3 2 3 2 2 3 0.008                       or x x x x x x  (viii)
  29. 29. 5. Find the base when: is the logarithamof 3 343 let base x   log 343 3 3 or x ,  343 3 3 Ans. HOME Solution: or x ,  7 ,  7  or x x (i)
  30. 30. 5. Find the base when: is the logarithamof 4 144 let base  x   log 144 4 4 or x ,  144 4 4 2 or x , 2 3     or x , 2 3 or x , 2 3 Ans. HOME Solution:         or x ,  2 3 , 2 3 4 4 4 4 4 2 2 4 4     or x x (ii)
  31. 31. 5. Find the base when: is the logarithamof let base  x    1 1 Ans. HOME Solution:   1 3 1 1     1 or x or , x or x ,  3 , 27 3 , 3 3 log 1 3 3  3 3 3          or x x (iii)
  32. 32. 5. Find the base when: let base  x    Ans. HOME Solution:   1 or , x a or x a a or x a a is the logarithamof x      , 1 , 1 1 log 1 1 1 1 (iv)
  33. 33. 6. Find the simplest value of : log 5 log 27 log 27 10 log 3 3log 3 3 25 log 5 10 log 5 10 log 5   Ans. HOME Solution: 3 2 or, 10 2log 5 10 log 3 or, log 5 log 3 or, log 25 log 3 or, 10 10 2 10 3 10 10 10 10  (i) 
  34. 34. 6. Find the simplest value of : log 27 if log 3 8 2 log 27 10 log 8 10 log 3 log 2 3log 3 10 3log 2 or, or, or, or, log 3 Ans. HOME Solution: a a   a  or, log 3 2 2 10 3 10 3 10  (ii)
  35. 35. 6. log log log 1 2 1 x x log Ans. HOME Find the simplest value of : (iii) x x a log if log Solution: 2 2 2  1 log x log x  x 2 1 2 3 2 2 1 1 2 1 log log log log 2 or, log 2 or, log 2 or, log 2 2 or, log 2 2 or, x x   1 1 1 3 or, 3 or, log 3 or, log 2 3 or, 3log 2 or, 3log 2 or, log 2 or, 2 2 3 a a x x x    
  36. 36. 7. Prove that log(1 23)  log1 log2  log3 proved HOME Solution: L H S . . . log(1  2  3) log(6) log(1  2  3) log1  log 2  log 3 or, or, or,
  37. 37. 8. Express M in term of N M N log 3log 1 2 1   M N   M N or, log  log  1   or, log log 3 log 1 solved HOME Solution: given :   3 3 1 1 1 2 3 M N or,   3 2 2 3 3   3 2 6 1 2 3 2 3 3 3 3 2 3 3 3 3 9 or, 3 or, 3 or, log 3log 1 2 1     N N M N M N M M N a a              (i)
  38. 38. 8. Express M in term of N (ii)   N M log 3 2log 10 10 N   M N M or, log  log  3 N M or, log (  )  3 N M or, log (  )  3  log 10 or, log ( ) log 10 2 3 or,   10 or,   1000 1000 solved HOME Solution: given : N M N M N M N M N M 1000 or, l og 3 log 2 2 3 10 2 10 10 2 10 2 10 2 10 10 2 10 10     
  39. 39. 9. Prove that : log log x a  x let y a x y a   log log . ..(i) log x x a a a log log a a x a     log 1 [ log 1] a a y x log log a a y x x a x proved HOME Solution: or, log or, a f rom (i) a a a      (i)
  40. 40. 9. Prove that : a 2log 2 2log x  a let y x a y x   log log . ..(i) 2log a a x x x 2log log x x a x     2log 1 [ log 1] x x y a log 2log x x y a or, log log x x 2 y a a 2log 2 x a proved HOME Solution: or, 2 or, x f rom (i) x x x       (ii)
  41. 41. 9. y x log log log y x y (iii)  z x   log log . ..(i) log y a a a y x log log a a log log   z y log log a a a a log x a z y y x log log proved HOME Prove that : Solution: or, log x or, a a f rom (i) a a a a x y x y let z x    
  42. 42. 9. m n n m (iv)    log log log log a b a b n b m b L H S m  n a b m a n a   n m R H S proved HOME Prove that : Solution: log log log log . . . log log log log log log or, or, or, log  log  . . . a b
  43. 43. 9. log 3  log 2  1 1 2 3 . . . log 3 log 2 proved HOME Prove that : Solution: 2 3 or, 1 . . . ] 1 log [ log log 3 or, log 3 2 2 R H S a b L H S b a      (v)
  44. 44. 9. b c a log  log  log  1 a b c . . . b c a log  log  log log c proved HOME Prove that : Solution: log b or, 1 . . . ] log log [ log log log log log or, R H S b a b a c b a L H S a a b c      (vi)
  45. 45. 9. Prove that :     2 2 log log log( ) log . . . x y log log x   x y x y a b a b a b or, log  log log  log [   (  )(  )] x   proved HOME Solution:        or, log( )log . . . 2 2 2 2 R H S y xy L H S y x y xy                  (vii)
  46. 46. 10. Find the value of : (i)  log 729 9 (27) 4 3 log 729 9 (27) 4 4  or, log 3 3 (3 ) 4 6 3  2  4 3 or, log 3 3 3 or, log 3 3 4 6 2 3 or, log 3 or, log 3 or, log 3 4 6 4 solved HOME Solution: given :         or, 1 3 4 3 4 3 6 3 4 3 3 6 2 1 3 3 3 1 3 4 3 3 1 3           
  47. 47. 10. Find the value of : log log log 81 3 2 3 log log log 81 3 2 3 or, log log log 3 [  3  81] or, log log 8log 3 [ log 1] 3 2 3 or, log log 2 or, log 3 solved HOME Solution: given :       or, 1 3 3 3 2 8 8 3 2 3  a a  (ii)
  48. 48. 10. Find the value of : log 16  log 9 2 3 log 16 log 9 2 3 or, log 2  log 3 [  2  16 3  9] or, 8log 2 4log 3 [ log 1] solved HOME Solution: given :             8 4 or, or, 2 2 3 4 8 4 3 8 2    a and a  (iii)
  49. 49. 10. Find the value of : b c d a (iv)    log log log log a b c d b c d a log  log  log  log a b c d log b solved HOME Solution: given : or, 1 log log log d log log log log or, a d c c b a   
  50. 50. 10. Find the value of : (v)   log 27 log 8 log 1000 log1.2 log 27  log 8  log 1000 3 3 3 log 3  log 2  log 10 12 3     log 3 log 2 log10 3 3 log10 2    log 3 3log 2 2     log 3  2log 2  log10   log 3  2log 2  log10 solved HOME Solution: given :     3 3     3 2 or, log 3 2log 2 log10 2 or, log 3 2log 2 log10 2 or, log 3 2 log10 3 or, log12 log10 or, 10 log or, log1.2 2 2 2 3  
  51. 51. 10. Find the value of : (vi)     log 4 log 5 log 6 log 7 log 3 3 4 5 6 7 log 4  log 5  log 6  log 7  log 3 log 4 solved HOME Solution: given : or, 1 log 3 log 7 log 7 log 6 log 6 log 5 log 5 log 4 log 3 or, 3 4 5 6 7    
  52. 52. 10. Find the value of : log 6 6 6... x let  6 6 6...  . ..(i) 2 x or, 6 6 6 6...   x x or,  6 from (i) or, 6    log 6 6 6... log 6 6 solved HOME Solution: given : or, log 6 1 6 2 6    x x (vii)
  53. 53. 11. 25     25    16 12 7  5 25 5 16 16  16 2 2 25 16 . . .  24     64           or, log 2 5 3 81 81   3 3  81 81  28 7        proved HOME Prove that: Solution:     or, log 2 5 1 or, log10 or, 1 . . . 2 5 2 3 3 5 or, log 2 2 5 2 3 3 5 or, log 2 80 24 15 or, log 2 80 log 24 log 15 or, log 2 log 80 7 log 24 12log 15 log 2 16log 1 80 7 log 24 12log 15 log 2 16log 1 64 36 28 24 16 7 28 16 12 28 36 12 16 16 7 4 4 12 3 2 16 4 16 12 7 R H S L H S                                                                           (i)
  54. 54. 11.      7 2 3 25  25        7 2 5 5 4  6 2 2 3 10  10    5 2  7 7 5 2 7 7 5 2 25 25 10 . . . 10    12 12  7 4 3 7 6 12 2 12 14        or, log 5 2 3 81 81 81 3 3   proved HOME Prove that: Solution:    0 1 0    or, log 5 2 3 or, log 1  2  1 or, log2 . . . 81  2 5 5 3 or, log 2 5 2 3 3 or, log 3  2 5 2 3 3 or, log 80 24 9 or, log 80 log 24 log 9 or, log 80 3log 24 2log 9 7 log log 2 80 3log 24 2log 9 7 log 1 12 3 4 14 12 3 6 2 14 3 4 4 2 3 2 7 2 3 R H S L H S                                                                              (ii)
  55. 55. 11.      7 5 3 25  25     10 5 16  16 2 2 5 81 81 81 25 25 16 . . . 16     12    28      28 15 12 10 7 3 12 7 5       or, log 2 5 3 proved HOME Prove that: Solution: 3 81       or, log 2  5  3 or, log 2  1  1 or, log2 . . . 3 2 5 2 3 3 5 or, log 2 5 2 3 3 5 or, log 80 24 15 or, log 80 log 24 log 15 or, log 80 3log 24 5log 15 7log log 2 80 3log 24 5log 15 7log 1 0 0 12 3 15 5 7 7 3 4 4 5 3 2 7 4 7 5 3 R H S L H S                                                                        (iii)
  56. 56. 11. 32    32  5 2 2 2         5 3 75 5 5    3 5 3 5 3 5 5 75 . . . 75    5 5 2 4  2 5 2   1 4 5 2 2 5 4     or, log 3 5 2 2 32    proved HOME Prove that: Solution:       or, log 3  5  2 or, log 1  1  2 or, log2 . . . 3 5 2 or, log 3 3 2 or, log 3 3 2 or, log 243 log 9 log 16 or, log 243 log 9 2log 16 log log 2 243 log 9 2log 16 log 0 0 1 5 2 4 2 5 4 4 2 5 4 2 R H S L H S                                                    (iv)
  57. 57. 11. (v) x y z y z z x x y log  log log  log log  log    y z z x x y log  log log  log log  log let k  x  y  z k x y z or, log log 1 y z z x x y log  log log  log log  log    y z z x x y log  log log  log log  log k x y z or, log  log  log  log k y z x z x y x y z or, log  log  log log  log  log log  log  log log k y x z x z y x y x z y z or, log  log log  log log  log log  log log  log log  log log proved HOME Prove that: Solution:         k or, log  0 k or, log log1 k or, 1 y z z x x y log log log log log log or,    1      x y z
  58. 58. 11. Prove that: 1    xy yz zx 1 1 . . . 1 xyz xyz xyz xy yz zx    xy yz zx or, log ( ) xyz xyz or, log ( ) 1 log ( ) xyz or, 2log log 1 proved HOME Solution:   or, 2 log 1 log or, log log log 1 log ( ) log ( ) 2 log ( ) log ( ) log ( ) 2            xyz a a b xy yz zx L H S xyz xyz xyz xyz a b a xyz xyz xyz   (vi)
  59. 59. 11. a b c    3 3 3 b c a . . . a b c log log log 3 3 3 c b    1 log log b proved HOME Prove that: Solution: log a log 1 1 27 or, 3 a 1 3 3 or, log log c 3log 3log 3log or, log log log log log log or, 1 27 log log log 3 3 3             a c b a b a a c b L H S b b c a  (vii)
  60. 60. 11. . . . 2 3 a a a a log  log  log  ...  log n a a a n a or, log  2log  3log  ...  log or, 1  2  3  ...  log proved HOME Prove that: Solution:             (  1)       ( 1) 2 log 2 or, log 2 1 log log log ... log 2 3 n n a sum of natural number n n n a L H S a n n a a a a n  (viii)
  61. 61. 11. x y z x y z (ix)      log log log log log log a b c b c a proved HOME Prove that: Solution: z c z a . . . x y z log  log  log a b c y   b y   c x a x b log log log log log log x y z L H S log log log or, log log log b c a or, log log log or,  
  62. 62. 11. Prove that: x y z log  log  log   1 1 1 1 y z x . . . x y z log  log  log 1 1 1 log y   log y log 1 x x 1 log log log proved HOME Solution: 1 or, 1 1 1 1 or, z log z log 1log log 1log 1log or, log log log or, 1 log 1 log 1 log or, 1 1 1                 x z z y y x x z y x z y L H S y z x (x)
  63. 63. 11. Prove that: x y z    2 2 2 x y z . . . x y z log log log 2 2 2 x y z    1 1 1 1   proved HOME Solution:    1 1 8 or, log 1 2 2 2 or, 1 1 1 2log 1 1 2log 2log or, 1 log log log log log or, 8 log log log 2 2 2           a x y z b a x y z L H S a x y z a b x y z   (xi)
  64. 64. 12 (a) log 25 8 log 25 log 8 log 5 2 log 2 3 2log 5 3log 2 10 2 2log  3log 2  2 log10 log 2 3log 2 2 1  0.3010 2 0.6990 solved HOME log 2 0.3010 , log 25. 8 If  find the valueof       0.9030 1.3980 0.9030 1.548 log 2 0.3010 given 3 0.3010 Solution:                
  65. 65. 12 (b) log 3 , log 5 , log 8. 30 30 30 If  a and  b find the valueof  30  log 8 log 2 3log 2  30 30               3 log 30 log 15 30 30    3 log 30 log 3 5 30 30    31 log 3 log 5 30 30    31 log 3 log 5    a  b    3 1 15 3log 30 30 30 3 30 solved HOME  log 3 and log 5 given 30 30   a  b Solution:
  66. 66. 13. (i) 1 1   If a 2  b  2  7 ab , show that, log a  b  log a  log b a  b  ab 2 2 a b ab ab ab or,   2  7  2 2     a b ab or,   9       ab ab  a b 1 1 1         a b ab 1  a b  a b a b ab a b ab a b log log 2 1 1 ( ) 3 3 or, or, log log 2 ( ) 3 or, log log 3 or, log 3 or, 9 or, 7 2 2 2 2 2 2                    2 proved HOME 3    Solution: given : [ Adding 2ab on both sides ] [ a2+b2+2ab=(a+b)2 ] [ taking log on both sides]
  67. 67. 13. (ii) 1 y x log log  , 23 2  log             1   2 2 25  2 or, 2 2 x y xy xy or,   2  25 2 2 x y xy xy or,   25  2 2 2 x y xy or,   23 y x or,   23 y x or, 23 log 5 or, log log log 5 or, 2log log log 2 5 log 2 2 2             x y xy xy xy x y xy xy x y x y x y x y x y proved HOME 5 x y x y showthat x y If Solution: given :
  68. 68. 13. (iii)   If a b a b showthat x x x x x , log log 3 5 5 3   x x x x 3  5 5  3    x x 5 3 5 3   x x x x 5  3 5   (3  )  x x x x a a b a b a b b a 5 3 5 3     b a x x 2 2 2 b a or, or, or, or, or, or, or, 2  2 2 2 2         or, log log b a b x a b a a x b b a a a a a a b x x x x x x   or, log log               a b a    proved HOME      Solution: given :
  69. 69. 13. (iv) 4 4 2 2 a  b  14 a b 4 4 2 2 2 2 2 2 a b a b a b a b or,   2  14  2       2 2 2 2 2 a b a b or,   16 2 2 2 2 a b ab or,   4 2 2 a b ab or,   4  a 2 b 2   ab        or, log   log 4 e e 2 2 a b a b or, log   log 4  log  log e e e e 2 2 2 a b a b or, log   log 2  log  log e e e e 2 2 a b a b or, log   2log 2  log  log e e e e or, log  a 2  b 2   log a  log b  2log 2 e e e e proved HOME 14 , log   log log 2log 2 4 4 2 2 2 2 e e e e If a  b  a b showthat a  b  a  b  Solution: given : [ Adding 2a2b2 on both sides ] [ a4+b4+2a2b2 =(a2+b2)2 ] [ taking log on both sides]
  70. 70. 14. (a) y y x x log log log       x k y z    z z log . ... .(i) y k z x log   . ... .(ii) z k x y log . ... .(iii)         from (i), (ii) and (iii) : x y z k y z k z x k x y l og log log           or, log x  y  z  ky  kz  kz  kx  kx  ky xyz or, log( )  0 or, log( ) log1 or,  1        xyz xyz k x y z x y z let HOME , 1 log log log       prove that xyz x y z x y z If Solution: proved [ log1=0]
  71. 71. 14. (b) (i) x y x log log log y z z             x k b c a x ak b c log log . ... .(i)       y k c a b y bk c a log    log   . ... .(ii) z k a b c z ck a b log    log   . ... .(iii)       from (i), (ii) and (iii) : a log x  b log y  c log z  ak b  c  bk c  a  ck a  b a b c x  y  z  akb  akc  bkc  bka  cka  ckb or, log log log a b c x y z or, log(   )  0 a b c x y z or, log( ) log1 or,  1        a b c x y z k a b c a b c let HOME , 1 log log log       a b c provethat x y z a b c a b c If Solution: proved [ log1=0]
  72. 72. 14. (b) (ii) y x y x log log log  z z          2 2                      x k b c b c x b c k b c k b c           log log . ... .(i) 2 2 y k c a c a y c a k c a k c a log     log      . ... .(ii) 2 2 z k a b a b z a b k a b k a b log     log      . ... .(iii) from (i), (ii) and (iii) :  b  c  log x   c  a  log y   a  b  log z  k  b  c   k  c  a   k  a  b   b  c   c  a   a  b                  x y z kb kc kc ka ka kb or, log log log         b  c c  a a  b x y z or, log    0 b  c c  a a  b x y z or, log log1       or, 1 2 2 2 2 2 2 2 2 2 2 2 2        b  c c  a a  b x y z k a b c a b c let HOME , 1 log log log       bc ca ab provethat x y z a b c a b c If Solution: [ log1 = 0] proved
  73. 73. 14. (b) (iii) y x y x log log log z z             x k ry qz p x pk ry qz       log log . ... .(i) y k pz rx q y qk pz rx log    log   . ... .(ii) z k qx py r z rk qx py log    log   . ... .(iii)       from (i), (ii) and (iii) : p log x  q log y  r log z  pk ry  qz  qk pz  rx  rk qx  py p q r x y z kpry kpqz kqpz kqrx krqx krpy or, log log log          x p y q z r    or, log    0 p q r x y z or, log log1 or,  1        p q r x y z k qx py pz rx ry qz let HOME , 1 log log log       p q r prove that x y z qx py pz rx ry qz If Solution: [ log1 = 0] proved
  74. 74. 15. log 45 2   log(3 5) 2   log 3 log 5   2log 3 log 5 10      2log 3 log10 log 2     2 0.4771 1 0.3010   0.9542 0.6990 1.6532 2 2log 3 log        HOME If log 2  0.3010, log3  0.4771, log 7  0.8451, f ind the value of : Solution: (i) solved [ log10 = 1]
  75. 75. 15. log108 2 3   log(2 3 ) 2 3   log 2 log 3   2log 2 3log 3     2 0.3010 3 0.4771   0.6020 1.4313 2.0333  HOME If log 2  0.3010, log3  0.4771, log 7  0.8451, f ind the value of : Solution: (ii) solved
  76. 76. 15. log 84 2    log(2 3 7) 2    log 2 log 3 log 7    2log 2 log 3 log 7     2 0.3010 0.4771 0.8451     0.6020 0.4771 0.8451 1.9242  HOME If log 2  0.3010, log3  0.4771, log 7  0.8451, f ind the value of : Solution: (iii) solved
  77. 77. 15. log 294    log(2 3 7 )    log 2 log 3 log 7    log 2 log 3 2log 7     0.3010 0.4771 2 0.8451     0.3010 0.4771 1.6902 2.4683 2 2  HOME If log 2  0.3010, log3  0.4771, log 7  0.8451, f ind the value of : Solution: (iv) solved
  78. 78. 15. log 21.6 216 10 log     log 216 log10  3 3     log 2 3 log10 3 3    log 2 log 3 log10    3log 2 3log 3 log10      3 0.3010 3 0.4771 1     0.9030 1.4313 1 1.3343       HOME If log 2  0.3010, log3  0.4771, log 7  0.8451, f ind the value of : Solution: (v) solved [ log10 = 1]
  79. 79. 16 (i) given If three positive real numbers a b and c are inG P , . ., solved showthat a b and c are in A P a, b and c are in G.P. c b   b c   log log            b a c b log  log  log  log a b c A P. a b b a hence, log , log and log are in . HOME log , log log . . Solution:
  80. 80. 16 (ii) log log x k log   log log log    log . ... .(i) y k log  2 . ... .(ii) z k log  3 . ... .(iii) from (i), (ii) and (iii) : y x k k k log  log  2   . ... .(iv) z y k k k log log 3 2 . ... .(v) from (iv) and (v) : y x z y     log log log log y z z    Hence, x, y and z are in G.P. y y x y x k x y z let                 log log 3 2 1 HOME , , , . . 3 2 1 provethat x y z are inG P x y z If   Solution:
  81. 81. 17. The f irst and the last terms of aG P are a and k respectively If the number of termbe n provethat k a log log given First term = a Last term = k No. of term = n Common ration = r . . .  Last terminG P ar 1 k ar log  log k a r log  log  log k a n r log  log   1 log k a n r log log 1 log proved HOME . log 1 , , where r is the commonratio r n    Solution:          n k a log log k a log log r n r k ar n n n n             1 log 1 log . . 1 1 1
  82. 82. 18. If the p q and r terms of aG P are a b and c respectively showthat th th th given , . . , , q  r a  r  p b  p  q c  pth term = a qth term = b rth term = c let the first term = b and common ratio = R pth term = a = b Rp-1 qth term = b = b Rq-1 rth term = c = b Rr-1 Press ( ) log ( ) log ( ) log 0 Solution:
  83. 83. since a, b and c are in G.P. - - - (ii) - - - (iii) - - - (i) b q p      R 1 1 1 R  1 q 1 1 1 1 1 1 R b c b b   and b c b R R R c b R a R R R a b a r r q r q r q q q p q p p                          b b  Press
  84. 84. from (i), (ii) and (iii) : 1 1 b q p c r q    c       q  p q p b r q c b  b     r  q q  p r  q q  p b  b  a  c r  q  q  p r  q q  p b  a  c r  p r  q q  p b  a  c r p r q q p r q r q q p b a c a b a b a                   log log log Press
  85. 85. r p b r q a q p c (  ) log  (  ) log  (  ) log r p b q r a p q c (  ) log   (  ) log  (  ) log q r a p q c r p b (  ) log  (  ) log  (  ) log  0 q r a r p b p q c (  ) log  (  ) log  (  ) log  0 solved HOME
  86. 86. 19. (i) If x  bc y  ca and z  ab showthat a b c 1 x y 1 z 1 x y z 1 1 bc ca ab a b c bc a ca b ab c a a b b c c 1 1 1 1 1 . .    abc abc abc a b c a b c    log log log log 1 1 1 1 log ( ) log ( ) log ( ) 1 log ( ) log log ( ) log log ( ) log log ( ) 1 log ( ) 1 log ( ) 1 1 1 1 1 1 1                    abc abc abc abc L H S abc proved HOME 1 1 ( ) log ( ), log ( ) log ,       i Solution:
  87. 87. 19. (ii) given If x  log ( bc ), y  log ( ca ) and z  log ab , showthat a b c ( ) 2 x bc log ( ) a  y ca log ( ) b  z ab log ( ) c  hence 1 1 1 bc ca ab a b c x y z x y z y x    1 1 x y     1 1 ( 1) ( 1) x y z z xy x y (   2)(  1)  (    1) xz  x  yz  y  z   xyz  xz  yz  z x  y  2 z  2  xyz  z x  y  2 z  z  2  xyz x y z xyz z z xy x y z z x y                                2 2 2 2 ( 1) ( 1)( 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 log 1 log 1 log 1 proved HOME ii x  y  z   xyz
  88. 88. 20. given a a a . . . 1     log log 2 log 3 1 a a a 2 3 4 a     a L H S log log log 2   a a log log 4 a log 4 a a log( 4 ) log 4 a log(2 ) a a a 2log 2 log 4 log 2 a       a a log 3 log 2 log 3 log 4 3    2 log 2 log 3 proved x a a 2  y log 2 a a 3  z log 3 a a log 4  yz a a a a a a a a a a a a a xyz a a a 2 log 4 2 log 3 2 log 4 1 log 4 1 log 4 log 3 log 2 3 4 3 2       HOME If x a y a and z a showthat xyz yz a a a log , log 2 log 3 , 1 2 2 3 4     
  89. 89. 21. log  , log  , prove log  p q p  given p q 1 p q . . . 1 x log log log x x 1 ab proved x  a x b p q  log log L H S p 1 1 1 1 a b b a ab ab b a x x x p q q            1 log log 1 log HOME b a If x a and x b that x q
  90. 90.   If log x y a, and log , log log 2 3    22. Solution: given: x log( x 2 y 3 ) a - - - (i) b x   log - - - (ii) y         solved (i) from 2 3 x y a log log   x y a 2log  3log  - - - (iii) x y b log log   log x  log y  b - - - (iv) from (iii) and (iv)   y b y a 2 log 3log    y b y a 2log 2 3log    y a b 5log 2 2 5 log (ii)   a b y from    and from a 2 b b 2 5 3 a b 2 3 5 log 5  log  5 5 5 log (iv) a b y a b x a b b x            HOME   b prove that x and y in terms of a and b y     
  91. 91. 23. e y 2  2  solved y y e  e 1 If x  y  y e y y e e   y y y y e e   y y e e e e y  y y  y  e  e  e  e e e y y y e y y y y e e   y y e e y y y y  likewise   e  e  e  e y y 1 - - - (ii) e 2 1 1 - - - (i) 2 1 1 e e y y y e e x e e x x                         y y y y     1 1 e e x x from and x y y e e e e x x y e y x x x e x e e e e e x e e y e e y y y y y y   1 1                    1 1 1 1 1 log 2 log 2 2 log 2 1 log log 1 log 2 2 1 (i) (ii) 2 2 HOME x x show that y e e      1 log 2 log , Solution: given:
  92. 92. 24. log 3 10 Showthat the valueof lies between and Solution: HOME 2 . 5 1 2 9 10 log 9 log 10 10 10 - - - (i) log 3  1 2log 3 1 1 2 log 3 10 10 2 10      and 243 100 log 243 log 100 10 10 log 3  log 10 5log 3  2log 10 10 10 - - - (ii) 5log 3  2 2 5 10 log 3 10 2 10 5 10     from i and ii 2 5 log 3 1 2 ( ) ( ) 10   proved
  93. 93. 25. Solution: HOME log 10 log (32 ) 5, . 6 If b and b find the value of a a a   a  log 10 (given) 10 a b log (32 ) 5 (given) 5 6 a b (6 )  32 - - - (ii) from (i) and (ii) 5 10  b a (6 ) 32  6 2 5 6 2   3 1 1 2 10 5 5 10 5 a 6 32 6 - - - (i) 5 5 5 5 5 5 a             a a a a b a b and b a found
  94. 94. 26. Solution: HOME found
  95. 95. 27. (i) Solve Solution: HOME 4 solved x x x 2 10 log  log  10 10 log 2 x  x   10 1 x x x x x x  x x x x x 2 2 2 10 1 2 10 10   1 2 1 10 2 10 10 10 2 10 10 10 log log log log log log log log log log log                                    2 2 2 1 x x log  log  4 10 10 x (log ) 4 x log  4   2 1  or , 10 x 2 hence x or 10 10 100 100 log log log log log 2 2 2 10 2 10 10 10 10 10 10  x or x x x x x          
  96. 96. 27. (ii) Solve Solution: log log log 1 2 2 2 x   x log log log 1 2 2 2   log log log log 2 2 2 2 2 log log 2 2 2  x log log  2  log 2 2 2 2 x log log  log 2 log log log 4 2 2 2 HOME solved log 4  2  2 4 , 16 2 2 2 2     or x x x x again x x
  97. 97. 27. (iii) Solve Solution: HOME log log log 11 8 4 2 x  x  x  log  log  log  11 8 4 2 1     x x x 1    x x x    1 1 1     2 3 6 6 1 1  solved 1 1 1 1 x 1 x 1 x 1    x log 6 , 64 2 6 log 2 11 11 log 2 11 6 log 2 1 11 2 3 log 2 11 1 log 2 2log 2 3log 2 11 log 2 log 2 log 2 11 log 2 log 4 log 8 6 2 3 2             or x x x x x x x x x 
  98. 98. 28. If a b and c are three consecutive positive integers showthat log(1 ) 2log Solution: HOME , ,   a b and c are three consecutive positiveintegers proved ac b b ac   1  , 2 2 b ac log  log(1  ) b ac 2log  log(1  ) 

×