SOLUTION: If 3^x = 5^y = (75)^z, show that z = xy/(2x + y)

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Question 1135275: If 3^x = 5^y = (75)^z, show that z = xy/(2x + y)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

If 3%5Ex+=+5%5Ey+=+75%5Ez, show that z+=+xy%2F%282x+%2B+y%29+

Let 3%5Ex+=+5%5Ey+=+75%5Ez+=+k.....1)
So, 3+=+k%5E%281%2Fx%29, 5+=+k%5E%281%2Fy%29 and 75+=+k%5E%281%2Fz%29
since 75+=+3+%2A+25+=+3+%2A+5%5E2
substituting the values from step 1) above,
k%5E%281%2Fz%29+=+%28k%5E%281%2Fx%29%29%2A%28k%5E%282%2Fy%29%29+=+k%5E%281%2Fx+%2B+2%2Fy%29 [By the law of exponents]
Equating the powers from both sides,
1%2Fz+=+1%2Fx+%2B+2%2Fy+=+%28y+%2B+2x%29%2Fxy
Taking reciprocal, z+=+%28x%2Ay%29%2F%282x+%2B+y%29 [Proved]