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Show that a^(log(a)x) = x....

Show that `a^(log_(a)x) = x`.

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answer any three questions: 3.if x>0, then show that log(1+x) > x/(1+x).

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Show that y= log (1+x) -(2x)/(2+x) , x gt -1 is an increasing function of xthroughout its domain.

If x gt 0 , show that, x gt log (1+x) gt x-x^(2)/2 .

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

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CALCUTTA BOOK HOUSE-LOGARITHM-Exercise - 7 (Long-answer type questions)
  1. Show that a^(log(a)x) = x.

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  2. Calculate : (i) log(2)root(4)(64root(3)(4^(-1)(8)^(-4/3)))

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  3. Calculate : (ii) log(5)sqrt(5sqrt(5sqrt(5....................oo)))

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  4. Calculate : (iii) log(3)(sqrt6)+log(3)(sqrt(2/3))-log(3)log(3)(9)

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  5. Calculate : (iv) 2log(2)sqrt(2sqrt(2sqrt(2....................oo)))

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  6. Calculate : (v) 1/6 sqrt((3log 1728)/(1+1/2 log 0.36 + 1/3log8))

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  7. Calculate : (vi) log(2)5 xx log(6)16 xx log(5) 8 xx log(8)6.

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  8. Simplify : (i) log(1/(sqrtx) )(y) xx log(1/(root(3)(y)))(z) xx log(1...

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  9. Simplify : (ii) 23 log 16/15 + 17log 25/24 + 10log 81/80

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  10. Simplify : (iii) 3log((36)/(25))+log((6)/(27))^(3)-2log((16)/(125))

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  11. Prove that log2+16log (16/15)+12log(25/24)+7log(81/80)=1

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  12. If log(40)4 = a and log(40)5 = b, then show that log(40)16=4(1-a-b).

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  13. If log(6)15 = alpha, log(12)18= beta and log(25)24 = gamma, then prove...

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  14. If log(12)18 = x and log(24)54 = y, then show that xy + 5(x-y) = 1

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  15. If log(a)M = (logb M) xx P, then express P in terms of a and b.

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  16. If 1/2 log(3)M + 3log(3)N = 1, then express M in terms of N.

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  17. Prove that 1/(log2 pi) + 1/(log(6)pi) > 2.

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  18. Prove that the value of log(10)3 lies in between 1/2 and 2/5.

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

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

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

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