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Prove that : (i) log(1^(1/5)+32^(1/5)+...

Prove that :
(i) `log(1^(1/5)+32^(1/5)+243^(1/5))=1/5(log1 + log 32 + log 243)`.

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CALCUTTA BOOK HOUSE-LOGARITHM-Exercise - 7 (Long-answer type questions)
  1. Prove that the value of log(10)3 lies in between 1/2 and 2/5.

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  2. Prove that the value of log(20)3 lies in between 1/2 and 1/3.

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  3. Prove that : (i) log(1^(1/5)+32^(1/5)+243^(1/5))=1/5(log1 + log 32 +...

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  4. Prove that : (ii) log(1+2+3) = log1 + log2 + log3.

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  5. Prove that : (iii) (yz)^(log(y/z))(zx)^(log(z/x))(xy)^(log(x/y)) = 1

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  6. Prove that : (iv) a^(log(a^2)x) xx b^(log(b^2)y) xx c^(log(c^2)z) = ...

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  7. Prove that : (v) p^(log(x)q) = q^(log(x)p)

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  8. Prove that : (vi) log(a)x + log(a^2)x^(2) + log(a^3)x^(3) + ………….+ l...

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  9. Prove that : (vii) log(sqrt a)b.log(sqrt(b))c.log(sqrt(c ))a = 8.

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  10. Prove that : (viii) (log(a)x)/(log(ab)x) = 1+log(a)b.

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  11. If x^(2)+y^(2)=z^(2), then prove that 1/(log(z-y)x) + 1/(log(z+y)x) = ...

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  12. If a = log(12)m and b = log(18)m, then prove that log(3)2= (a-2b)/(b-2...

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  13. If x^(2)+y^(2) = 6xy, then prove that 2log(x+y) = log x + logy + 3log2...

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  14. If log((x+y)/2) = 1/3{log x + logy + log(x+y)}, then prove that (x^2)/...

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  15. If a^(2-x).b^(5x) = a^(x+3).b^(3x), then show that x log (b/a) = 1/2 l...

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  16. If (logx)/(y-z) = (logy)/(z-x) = (logz)/(x-y), then prove that (i) x...

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  17. If (log x)/(y-z) = (log y)/(z-x) = (log z)/(x-y), then prove that xyz ...

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  18. If (logx)/(ry-qz) = (log y)/(pz - rx) = (log z)/(qx - py), then prove ...

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  19. If y = a^(1/(1-log(a)x)) and z = a^(1/(1-log(a)y)), then show that x =...

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  20. If x = log(c ) b + log(b)c , y=log(a)c + log(c ) a, z=log(b)a+log(a)b,...

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