Question

2^x = 3^y = 5^z = 30 ^ 1/x+1/y+1/z then find the value of x,y,z

Answer 1

Answer:

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Answer 2

Given,

$\[{2^x} = {3^y} = {5^z} = {30^{\left( {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right)}}\]$

Solution,

Consider, $\[{2^x} = {3^y} = {5^z} = k\]$

$\[\begin{array}{l}{2^x} = k\\ \Rightarrow x\log 2 = \log k\\ \Rightarrow x = \frac{{\log k}}{{\log 2}}\end{array}\]$

$\[\begin{array}{l}{3^y} = k\\ \Rightarrow y\log 3 = \log k\\ \Rightarrow y = \frac{{\log k}}{{\log 3}}\end{array}\]$

$\[\begin{array}{l}{5^z} = k\\ \Rightarrow z\log 5 = \log k\\ \Rightarrow z = \frac{{\log k}}{{\log 5}}\end{array}\]$

$\[{30^{\left( {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right)}} = k\]$

$\[ \Rightarrow {30^{\left( {\frac{{\log 2}}{{\log k}} + \frac{{\log 3}}{{\log k}} + \frac{{\log 5}}{{\log k}}} \right)}} = k\]$

$\[ \Rightarrow {30^{\left( {\frac{{\log 2 + \log 3 + \log 5}}{{\log k}}} \right)}} = k\]$

$\[ \Rightarrow {30^{\left( {\frac{{\log 30}}{{\log k}}} \right)}} = k\]$

$\[ \Rightarrow \left( {\frac{{\log 30}}{{\log k}}} \right)\log 30 = \log k\]$

$\[\begin{array}{l} \Rightarrow {\left( {\log 30} \right)^2} = {\left( {\log k} \right)^2}\\ \Rightarrow \log k = \log 30\\ \Rightarrow k = 30\end{array}\]$

Calculate the value of x.

$\[\begin{array}{l}{2^x} = k\\ \Rightarrow {2^x} = 30\\ \Rightarrow x\log 2 = \log 30\\ \Rightarrow x = \frac{{\log 30}}{{\log 2}}\end{array}\]$

Calculate the value of y.

$\[\begin{array}{l}{3^y} = k\\ \Rightarrow {3^y} = 30\\ \Rightarrow y\log 3 = \log 30\\ \Rightarrow y = \frac{{\log 30}}{{\log 3}}\end{array}\]$

Calculate the value of z.

$\[\begin{array}{l}{5^z} = k\\ \Rightarrow {5^z} = 30\\ \Rightarrow z\log 5 = \log 30\\ \Rightarrow z = \frac{{\log 30}}{{\log 5}}\end{array}\]$