求解8，9要有算式，會選最佳解乾溫?

( 8 - 1 )
pf :
a^x = (ab)^z = a^z * b^z
a^x / a^z = b^z
a^(x-z) = b^z
兩邊取對數得 :
(x-z) * log a = z * log b
log a / log b = z / (x-z) ..... (1式)

b^y = (ab)^z = a^z * b^z
b^y / b^z = a^z
b^(y-z) = a^z
(y-z) * log b = z * log a
log a / log b = (y-z) / z ..... (2式)

由 (1式) 與 (2式) 得 :
z / (x-z) = (y-z) / z
z² = (x-z)(y-z) = xy - xz - yz + z²
xy - xz - yz = 0
xy = xz + yz
等式兩邊同除以 xyz 得 :
1/z = 1/y + 1/x
Q.E.D.

( 8 - 2 )
令 a = 3 , b = 5 , 則 ab = 15
由第(1)小題知
1/x + 1/y = 1/z
等式兩邊同乘以 xyz 得 :
yz + xz = xy
xy - yz - zx = 0
Ans: 0

( 8 - 3 )
3^x = 5^y = 15^(z/3)
令 u = z/3 , 則
3^x = 5^y = 15^u
由第(1)小題知
1/x + 1/y = 1/u
1/x + 1/y = 1/(z/3) = 3/z
z/x + z/y = 3
Ans: 3

( 9 )
a^x = b^y = c^z = 30^ω
取對數得 :
x * log a = y * log b = z * log c = ω * log 30
因此 :
1/x = log a / ( ω * log 30 )
1/y = log b / ( ω * log 30 )
1/z = log c / ( ω * log 30 )

1/ω = 1/x + 1/y + 1/z = ( log a + log b + log c ) / ( ω * log 30 )
ω ( log a + log b + log c ) = ω * log 30
log a + log b + log c = log 30
log ( abc ) = log 30
abc = 30 = 2*3*5
又 a < b < c , 所以 :
a = 2 , b = 3 , c = 5 ..... Ans