Question 1135275

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


Let {{{3^x = 5^y = 75^z = k}}}.....1)

So, {{{3 = k^(1/x)}}}, {{{5 = k^(1/y)}}} and {{{75 = k^(1/z)}}}

since {{{75 = 3 * 25 = 3 * 5^2}}}

substituting the values from step 1) above,

{{{k^(1/z) = (k^(1/x))*(k^(2/y)) = k^(1/x + 2/y)}}} [By the law of exponents]

Equating the powers from both sides,

{{{1/z = 1/x + 2/y = (y + 2x)/xy}}}

Taking reciprocal, {{{z = (x*y)/(2x + y)}}} [Proved]