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Question

Given that

logb(yz^2) = m
logb(y^3 z) = n

express in terms of m and n:

logbz/y
logb squareroot yz


All are of log base b.
y^3 z is y-to-the-power-of-3 * z and not y-to-the-power-of-3z.

Answer 1

logb (yz^2) = m
logb y + logb z^2 = m
logb y + 2logb z = m ---(1)

logb (y^3z) = n
logb y^3 + logb z = n
3logb y + logb z = n ---(2)

2x(2)
6logb y + 2logb z = 2n---(3)
(3)-(1)
5logb y = 2n - m
logb y = 1/5 (2n - m) --(4)
Substitute into (1)
1/5 (2n - m) + 2logb z = m
2n - m + 10logb z = 5m
10logb z = 6m - 2n
logb z = 1/5 (3m - n) ---(5)

logb z/y
= logb z - logb y
= 1/5 (3m - n) - 1/5 (2n - m) ----(from (4) and (5))
= 4/5 m - 3/5 n
= 1/5(4m - 3n)

logb sqrt(yz)
= logb (yz)^(1/2)
= 1/2 logb (yz)
= 1/2 [logb y + logb z]
= 1/2 [1/5 (2n - m) + 1/5 (3m - n)]
= 1/2 (2/5 m + 1/5 n)
= 1/10 (2m + n)

Answer 2

I'll give you a hint:
- log(ab)=loga+logb for any base
- log(a/b)=loga-logb for any base
- log (x)^n=nlog(x) for any base