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Logarithm

Class 9th Mathematics West Bengal Board Solution
Let Us Do The Sum 21
  1. Let us evaluate: (i) log_4 (1/64) (ii) log0.01 0.000001 (iii) log_ root 6 216 (iv)…
  2. Let us write by calculating, find its base if logarithm of 625 is 4…
  3. Let us write by calculating, find its base if logarithm 5832 is 6…
  4. If 1 + log10a = 2log10b, then express a by b
  5. If 3 + log10x = 2log10y, then express x by y
  6. log2[log2{log3 (log327^3)}] Let us evaluate:
  7. logroot 27+log8-logroot 1000/log1.2 Let us evaluate:
  8. log34 × log45 × log56 × log67 × log73 Let us evaluate:
  9. log_10 384/5 + log_10 81/32 + 3log_10 5/3 + log_10 1/9 Let us evaluate:…
  10. log 75/16 - 2log 5/9 + log 32/243 = log2 Let us prove:
  11. log1015(1+log1530) + 1/2 log1016(1+ log47)- log106(log63+1+ log67) = 2 Let us prove:…
  12. log_2 log_2 log_4256+2log_ root 2 2 = 5 Let us prove:
  13. log_ x^2 x x log_ y^2 y x log_ z^2 z = 1/8 Let us prove:
  14. log_ b^3 a x log_ c^3 b x log_ a^3 c = 1/27 Let us prove:
  15. 1/log_xy (xyz) + 1/log_yz (xyz) + 1/log_zx (xyz) = 2 Let us prove:…
  16. log a^2/bc+log b^2/ca+log c^2/ab = 0 Let us prove:
  17. xlogy - logz × ylogz - logx × zlogx - logy = 1 Let us prove:
  18. If log x+y/5 = 1/2 (logx+logy) then let us show that x/y + y/z = 23…
  19. If a^4 + b^4 = 14a^2 b^2 , then let us show that log (a^2 + b^2) = log a + log b +…
  20. If logx/y-z = logy/z-x = logz/x-y then let us show that xyz = 1
  21. If logx/b-c = logy/c-a = logz/a-b then let us show that xyz = 1 (a)xb+c . yc+a . za+b =…
  22. If, a3-x . b5x = a5+x . b3x, then let us show that, xlog (b/a) = loga…
  23. log_8[log_2 log_3 (4^x + 17)] = 1/3 Let us solve:
  24. log8x + log4x + log2x = 11 Let us solve:
  25. Let us show that the value of log102 lies between 1/4 1/3
  26. If log_ root x 0.25 = 4 then the value of xA. 0.5 B. 0.25 C. 4 D. 16…
  27. If log_10 (7x-5) = 2 then the value of xA. 10 B. 12 C. 15 D. 18
  28. If log23 = a, then the value of log827 isA. 3a B. 1/a C. 2a D. a
  29. If log_ root 2 x = a then the value of log_ 2 root 7 x isA. a/3 B. a C. 2a D. 3a…
  30. If log_x 1/3 = - 1/3 then the value of x isA. 27 B. 9 C. 3 D. 1/27…
  31. Let us calculate the value of log4log4log4 256.
  32. Let us calculate the value of log a^n/b^n + log b^n/c^n + log c^n/a^n…
  33. Let us show that a^log_1^x = x
  34. If loge2 . logx25 = log1016 . loge10, then let us calculate the value of x.…
Let Us Do The Sum 21
Question 1.

Let us evaluate:

(i)

(ii) log0.01 0.000001

(iii)

(iv)


Answer:

(i)



As, logaMc = c logaM



(ii)



As, logaMc = c logaM



[As, logaa = 1]


(iii)



As, logaMc = c logaM



[As, logaa = 1]


(iv)



= log√3(√3)6


= 6(log√3√3) [As, logaMc = c logaM]


= 6(1) [As, logaa = 1]


= 6
Question 2.

Let us write by calculating, find its base if logarithm of 625 is 4
Answer:

Let the base be a.


We know that if


Then


So,





We know that if the powers are same on both the sides then the bases must also be same.


Therefore, a=5
Question 3.

Let us write by calculating, find its base if logarithm 5832 is 6
Answer:

Let the base be a.


We know that if


Then


So,
Question 4.

If 1 + log10a = 2log10b, then express a by b
Answer:

Given expression :


-
Question 5.

If 3 + log10x = 2log10y, then express x by y
Answer:

Given expression :


-
Question 6.

Let us evaluate:

log2[log2{log3 (log3273)}]
Answer:









Question 7.

Let us evaluate:
Answer:









Question 8.

Let us evaluate:

log34 × log45 × log56 × log67 × log73
Answer:

log34 × log45 × log56 × log67 × log73


= 1
Question 9.

Let us evaluate:
Answer:









Question 10.

Let us prove:
Answer:

LHS




RHS
Question 11.

Let us prove:

log1015(1+log1530) + log1016(1+ log47)– log106(log63+1+ log67) = 2
Answer:

LHS = log1015(1+log1530) + log1016(1+ log47)– log106(log63+1+ log67)
Question 12.

Let us prove:
Answer:

LHS





RHS
Question 13.

Let us prove:
Answer:

LHS




RHS
Question 14.

Let us prove:
Answer:

LHS



RHS
Question 15.

Let us prove:
Answer:

LHS






RHS
Question 16.

Let us prove:
Answer:

LHS



RHS
Question 17.

Let us prove:

xlogy – logz × ylogz – logx × zlogx – logy = 1
Answer:

LHS = xlogy – logz × ylogz – logx × zlogx – logy


Now, taking log we have,






Now, RHS = 1


Taking log we have, log 1= 0 =LHS


Therefore, LHS=RHS


Hence, proved.
Question 18.

If then let us show that


Answer:












Question 19.

If a4 + b4 = 14a2b2, then let us show that log (a2 + b2) = log a + log b + 2log2.
Answer:

Given that





…..eq(1)


LHS


(from eq(1))



= RHS


Therefore, LHS =RHS


Hence , proved.
Question 20.

If then let us show that xyz = 1
Answer:

Given that

Let


…..eq(1)


…..eq(2)


…..eq(3)


Now, adding eq(1) , eq(2) and eq(3)






Hence, proved.
Question 21.

If then let us show that xyz = 1

(a)xb+c . yc+a . za+b = 1

(b)
Answer:

Given that

Let


…..eq(1)


…..eq(2)


…..eq(3)


Now, adding eq(1) , eq(2) and eq(3)






(a) LHS = xb+c . yc+a . za+b


Taking log we have,









0


RHS


Therefore, xb+c . yc+a . za+b = 1


(b) LHS


Taking log we have,








0


RHS


Therefore,



Question 22.

If, a3–x . b5x = a5+x . b3x, then let us show that,


Answer:

Given that a3–x . b5x = a5+x . b3x

Taking log on both sides










Hence ,proved .
Question 23.

Let us solve:
Answer:














Therefore, x = 3.
Question 24.

Let us solve:

log8x + log4x + log2x = 11
Answer:










Question 25.

Let us show that the value of log102 lies between


Answer:

Let


Now, LCM of denominator of and










Question 26.

If then the value of x
A. 0.5

B. 0.25

C. 4

D. 16
Answer:


We know that if


Then


So,





We know that if the powers are same on both the sides then the bases must also be same.
Question 27.

If then the value of x
A. 10

B. 12

C. 15

D. 18
Answer:


We know that if


Then


So,
Question 28.

If log23 = a, then the value of log827 is
A. 3a

B.

C. 2a

D. a
Answer:

We know,





Now, we know that logaMc = c logaM



[As, logaa = 1]
Question 29.

If then the value of is
A.

B. a

C. 2a

D. 3a
Answer:

Since

We know that if


Then


So,


…….eq(1)


Now,






Question 30.

If then the value of x is
A. 27

B. 9

C. 3

D.


Answer:







Question 31.

Let us calculate the value of log4log4log4 256.
Answer:






Question 32.

Let us calculate the value of


Answer:





Question 33.

Let us show that


Answer:

LHS

Taking log we have,





RHS


Taking log we have,


Therefore, LHS = RHS


Hence , proved .
Question 34.

If loge2 . logx25 = log1016 . loge10, then let us calculate the value of x.
Answer:

Given that







Therefore,