Logarithms

Given:
log_a xy³ = m
log_a x³ y² = n
Express log_a (xy)^1/2

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log_a xy³ = log_a a^m
xy³ = a^m
x = a^m / y³ --------- 1

log_a x³ y² = n
x³ y² = aⁿ
x³ = aⁿ / y²
x = (aⁿ / y²) ^1/3 --------- 2

a^m / y³ = (aⁿ / y²) ^1/3
a^3m / y⁹ = aⁿ / y²
a^3m / aⁿ = y⁹ / y²
y⁷ = a^3m-n
y = (a^3m-n)^1/7

x = a^m / y³
x = a^m / [(a^3m-n)^1/7]³
x⁷ = a^7m / a^9m-3n
x⁷ = a^7m-9m+3n
x⁷ = a^3n-2m
x = (a^3n-2m)^1/7

log_a(xy)^1/2
= log_ax^1/2y^1/2
= 1/2 log_ax + 1/2 log_ay
= 1/2 log_a(a^3n-2m)^1/7 + 1/2 log_a(a^3m-n)^1/7
= 1/14 log_aa^3n-2m + 1/14 log_aa^3m-n
= 1/14 (3n-2m+3m-n)
= 1/14 (2n+m)

*SOLVED*
Level: High

you use replacement method didn't you , okay standard for me but still sort of high standard