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If a^(2-x).b^(5x) = a^(x+3).b^(3x), then...

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

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CALCUTTA BOOK HOUSE-LOGARITHM-Exercise - 7 (Long-answer type questions)
  1. If x^(2)+y^(2) = 6xy, then prove that 2log(x+y) = log x + logy + 3log2...

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

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

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

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

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

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  8. 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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  9. If a,b,c be three such positive numbers (none of them is 1) that (log(...

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  10. If x = log(2a)a, y = log(3a)2a , z = log(4a)3a, then show that xyz + 1...

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  11. If a > 0, c > 0, b = sqrt(ac), ac != 1 and N > 0, then prove that (log...

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  12. If (a(b+c-a))/(loga) = (b(c+a-b))/(logb) = (c(a+b-c))/(logc), then pro...

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  13. If log(a+b+c) = log a + log b + log c, then prove that log ((2a)/(1-...

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  14. If x+y = z, then prove that 1/(log((sqrt(z)-sqrt(y)))(x)) + 1/(log((sq...

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  15. If y = (e^(x)-e^(-x))/(e^(x)+e^(-x)) then prove that y = (e^(2x)-...

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  16. Solve : (i) x^(log(10)x)= 100x.

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  17. Solve : (ii) log(x)2log(x/16)2 = log(x/64)2.

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  18. Solve : (log(2)(x-4)+1)/(log(sqrt2)(sqrt(x+3)-sqrt(x-3))) = 1.

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  19. Solve : (iv) x^(log(2)a)+a^(log(2)x)=2a^(2).

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  20. Solve : (v) 2log(2)log(2)x+log(1/2)log(2)(2sqrt(2)x) = 1.

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