There are 3 number a, b and c such that ` log_(10) a = 5.71, log_(10) b = 6.23 and log_(10) c = 7.89`. Find the number of digits before dicimal in ` (ab^(2))/c`.
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A certain number of two digits is four times the sum of the digits. If 9 is added to the number the digits
Correct option is (C) 12 Let the two-digit number be ab whose unit digit is b and ten's digit is a. \(\therefore\) Required number is ab = 10a + b _________(1) Reversed number is...
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Find the value of the following: (i) ` log_(10) 2 + log_(10) 5` (ii) ` log_(3) (sqrt(11)-sqrt2) + log_(3) (sqrt11+sqrt2)` (iii) ` log_(7) 35 - log_(7)
(i) `log_(10)2+ log_(10) 5 = log_(10)(2 xx 5)= log_(10) 10 = 1` `(ii) log_(3)(sqrt11-sqrt2)+log_(3)(sqrt11+sqrt2)` ` " "= log_(3) (sqrt11-sqrt2) xx (sqrt11+sqrt2)` `" "= log_(3) (11-2)=log_(3)9=2 (as 3^(2) = 9)` `(iii)...
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Write each of the following as single logarithm: ` (a) 1+ log_(2) 5" "(b) 2- log_(3) 7` `(c) 2log_(10) x+3 log_(10) y - 5 log_(10) z`
Correct Answer - (a)`log_(2) 10" "(b) log_(3) 9/7" "(c) log_(10)(x^(2)y^(3))/z^(5)` (a) `1+ log_(2) 5= log_(2) 2+ log_(2) 5 = log_(2) (2 xx5) = log_(2) 10` (b) `2-log_(3) 7 = 2 log_(3)...
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Prove the following identities: (a) `(log_(a) n)/(log_(ab) n) = 1+ log_(a) b" "(b) log_(ab) x = (log_(a) x log_(b) x)/(log_(a) x + log_(b) x)`.
(a) `(log_(n) n)/(log_(ab) n) = (log_(n) ab)/(log_(n) a) = (log_(n) a+ log_(n) b)/(log_(n) a)` ` = 1+(log_(n) b)/(log_(n) a) = 1+log_(a) b` (b) `(log_(a) xlog_(b) x)/(log_(a) x+log_(b) x) =(1/(log_(x) a)1/(log_(x)...
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The value of x satisfying the equation `root(3)(5)^(log_(5)5^(log_(5)5^(log_(5)5^(log_(5)((x)/(2))`= 3, is
Correct Answer - D Using ` a^(log_(a)N) = N` repeatedly, we get `(root(3)5)^(log _(5)(x/2)) = 3` `rArr 5^(log_(5)(x/2)^(1//3)) = 3` `rArr (x/2)^(1//3)= 3` ` rArr x/2 = 27` `rArr x = 54`
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If `xy^(2) = 4 and log_(3) (log_(2) x) + log_(1//3) (log_(1//2) y)=1` , then x equals
Correct Answer - D `log_(3)(log_(2)x)+log_(1//3)(log_(1//2)y)= 1` `or log_(3)(log_(2)x)-log_(3)(log_(1//2)y) = 1` ` or log_(3)(log_(2)(4//y^(2)))-log_(3)(log_(1//2)y) = 1` ` or log_(2)(4//y^(2))=3(log_(1//2)y)` ` or log_(2)(4//y^(2))=- 3 (log_(2)y)` `or log_(2)(4//y^(2))+(log_(2)y^(3))=0` ` or 4y = 1` `...
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The value of ` (6 a^(log_(e)b)(log_(a^(2))b)(log_(b^(2))a))/(e^(log_(e)a*log_(e)b))` is
Correct Answer - A::B `(6a^(log_(e)b)log_(a^(2))b*log_(b^(2))a)/(e^(log_(e)a*log_(e)b))=(6a^(log_(e)b)1/2log_(a)b*1/2log_(b)a)/((e^(log_(e)a))^(log_(e)b))` ` (6/4 a^(log_(e)b))/(a^(log_(e)b))` `= 3/2`
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Consider the system of equations ` log_(3)(log_(2)x)+log_(1//3)(log_(1//2)y) =1 and xy^(2) = 9`. The value of x in the interval
Correct Answer - C `log_(3)(log_(2)x)+log_(1//3)(log_(1//2)y) = 1` ` rArr log_(3) (log_(2)x)-log_(3)(-log_(2)y) = 1` ` rArr log_(3)(-(log_(2)x)/(log_(2)y)) = 1` ` rArr -(log_(2)x)/(log_(2)y) = 3` ` rArr xy^(3) = 1` Also, ` xy^(2)...
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Consider the system of equations ` log_(3)(log_(2)x)+log_(1//3)(log_(1//2)y) =1 and xy^(2) = 9`. The value of 1/y lies in the interval
Correct Answer - B `log_(3)(log_(2)x)+log_(1//3)(log_(1//2)y) = 1` ` rArr log_(3) (log_(2)x)-log_(3)(-log_(2)y) = 1` ` rArr log_(3)(-(log_(2)x)/(log_(2)y)) = 1` ` rArr -(log_(2)x)/(log_(2)y) = 3` ` rArr xy^(3) = 1` Also, ` xy^(2)...
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If `x gt 0`, `y gt 0`, `z gt 0`, the least value of `x^(log_(e)y-log_(e)z)+y^(log_(e)z-log_(e)x)+Z^(log_(e)x-log_(e)y)` is
Correct Answer - A `(a)` Let `log_(e)x=a`, `log_(e)y=b`, `log_(e)z=c` `impliesx=e^(a)`, `y=e^(b)`, `z=e^(c )` So, given expression `e^(a(b-c))+e^(b(c-a))+e^((a-b))` Using A.M. ge G.M. `:.(e^(a(b-c))+e^(b(c-a))+e^(c(a-b)))/(3)` ge [a^(a(b-c)+b(c-a)+c(a-b))]^(1//3)` `:.e^(a(b-c))+e^(b(c-a)+e^(c(a-b)))ge 3`
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