Question

5^x=3^y=45^z. then prive that 1/z=1/x+2/y

Answer 1

your answer for above question

Answer 2

Hey user

Here is your answer :-

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${5}^{x} = {3}^{y} = {45}^{z} \\\ \\ let \: {5}^{x} = {3}^{y} = {45}^{z} = k \\ \\ {5}^{x} = k \\ \\ taking \: log \: on \: both \: the \: sides \: \\ \\ x log(5) = log(k) \\ \\ \frac{1}{x} = \frac{ log(5) }{ log(k) } = log_{k}(5) \\ \\ \frac{1}{x} = log_{k}(5) \: \: \: \: \: \: \: \: \: ...(1) \\ \\ similarly \\ \\ {3}^{y} = k \\ \\ y log(3) = log(k) \\ \\ \frac{1}{y} = \frac{ log(3) }{ log(k) } = log_{k}(3) \\ \\ \frac{1}{y} = log_{k}(3) \: \: \: \: \: \: \: ...(2) \\ \\ {45}^{z} = k \\ \\ zlog(45) = log(k) \\ \\ \frac{1}{z} = \frac{ log(45) }{ log(k) } = log_{k}(45) \\ \\ \frac{1}{z} = log_{k}(45) \: \: \: \: \: \: \: ...(3) \\ \\ now \: on \: lhs \: \\ \\ \frac{1}{z} = log_{k}(45) \: \: \: \: \: \: \: \: ...from \: (3) \\ \\ = log_{k}(9 \times 5) \\ \\ = log_{k}(9) + log_{k}(5) \\ \\ = log_{k}( {3}^{2} ) + log_{k}(5) \\ \\ = 2 log_{k}(3) + log_{k}(5) \\ \\ = 2( \frac{1}{y} ) + \frac{1}{x} \: \: \: \: \: \: \: \: \: ...from \: (1) \: and \: (2) \\ \\ = \frac{1}{x} + \frac{2}{y} \\ \\ hence \: proved$

Thank you.