log x4 & M1x6

1.
log(2x^3 ) - 5logx + log(5x^2 )
= log(2x^3 ) - logx^5 + log(5x^2 )
= log(2x^3*5x^2/x^5)
= log10
= 1


2.
(logx^2 - log√x) / (log x^3 + log√x)
= [2logx - (1/2)logx] / [3logx + (1/2)logx]
= [(3/2)logx] / [(7/2)logx]
= 3/7


3.
[log(x^3y^4) - log(xy^4)] / logx^3
= log(x^3y^4 / xy^4) / logx^3
= logx^2 / logx^3
= 2logx / 3logx
= 2/3


4.
(2logx^3 - 3log x^4) / [log(xy^2)- 2log(xy)]
= (6logx - 12logx) / (logx + 2logy - 2logx - 2logy)
= -6logx / -logx
= 6


1.
x^2 term = 7C5(a)^5(2x)^2 = 20412x^2
(7!/5!2!)(a^5)(4x^2) = 20412x^2
84a^5x^2 = 20412x^2
a^5 = 243
a = 3


2.
x^2 term = nCn-2[(1)^(n-2)](-4x)^2 = 336x^2
[n!/(n - 2)!2!](16x^2) = 336x^2
[n(n - 1)/2] = 21
n^2 - n - 42 = 0
(n - 7)(n + 6) = 0
n = 7 or n = -6 (rejected)
Hence, n = 7


3.
x^2 term = nCn-2[(1)^(n - 2)](-x/2)^2 = 7x^2
[n!/(n - 2)!2!][(x^2)/4] = 7x^2
n(n - 1) = 56
n^2 - n - 56 = 0
(n - 8)(n + 7) = 0
n = 8 or n = -7 (rejected)
Hence, n = 8


4.
x^2 term = nCn-2[(1)^(n - 2)](x/3)^2 = 4x^2
[n!/(n - 1)!2!][(x^2)/9] = 4x^2
n(n - 1) = 72
n^2 - n - 72 = 0
(n - 9)(n + 8) = 0
n = 9 or n = -8 (rejected)
Hence, n = 9


5.
General term = 8Cn(x^n)[(1/2x)^(8 - n)]
For the constant term: (x^n)[(1/x)^(8 - n)] = x^0
n - (8 - n) = 0
n = 4

Constant term
= 8C4(x^n)[(1/2x)^(4)]
= (8!/4!4!)(x^n)(1/16x^n)
= 70/16
= 35/8


6.
General term = 9Cn[(3x)^n][(1/x^2)^(9 - n)]
For constant term : (x^n)[(1/x^2)^(9 - n)] = x^0
n - 2(9 - n) = 0
n = 6

Constant term
= 9C6[(3x)^6][(1/x^2)^3]
= (9!/6!3!)(729x^6)(1/x^6)
= 84 * 729
= 61236