Question

prove that (question is given above in the picture)

Answer 1

HEYA !

Let
${2}^{x} = {5}^{y} = {10}^{z} = k \\ \\ \Rightarrow \: \: \: {2}^{x} = k \\ \\ \Rightarrow \: \: \: 2 = {k}^{ \frac{1}{x} } \: \: \: \: \: \: \: \: \: .....(1) \\ \\ similarly \\ \\ \Rightarrow \: \: \:5 = {k}^{ \frac{1}{y} } \: \: \: \: \: \: \: \: .....(2) \\ and \\ \Rightarrow \: \: \:10 = {k}^{ \frac{1}{z} } \: \: \: \: \: \: .....(3) \\ \\ \text{multiplying (1) and (2) we get} \\ \\ 2 \times 5 = {k}^{ \frac{1}{x} } \times {k}^{ \frac{1}{y} } \\ \\ \Rightarrow \: \: \:10 = {k}^{ (\frac{1}{x} + \frac{1}{y}) } \: \: \: \: \: .....(4) \\ \\ \text{from (3) and (4) we get} \\ \\ {k}^{ \frac{1}{z} } = {k}^{ \frac{1}{x} + \frac{1}{y} } \\ \\ \Rightarrow \: \: \: \boxed{\frac{1}{z} = \frac{1}{x} + \frac{1}{y} }$

Answer 2

Hope u like my process
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$given \: \: \: {2}^{x} = {5}^{y} = {10}^{z} \\ \\ let \: \: \: {2}^{x} = {5}^{y} = {10}^{z} = k \: (constant) \\ \\ so. \: \\ {2}^{x} = k \\ or. \: 2 = {k}^{ \frac{1}{x} } \\ \\ {5}^{y} = {k} \\ or. \: 5 = {k}^{ \frac{1}{y} } \\ \\ {10}^{z} = k \\ or. \: 10 = {k}^{ \frac{1}{z} } \\ \\ now... \\ 2 \times 5 = {k}^{ \frac{1}{x} } \times {k}^{ \frac{1}{y} } \\ or. \: 10 = {k}^{ \frac{1}{x} + \frac{1}{y} } \\ or. \: {k}^{ \frac{1}{z} } = {k}^{ \frac{1}{x} + \frac{1}{y} } \\ \\ so \\ \frac{1}{x} + \frac{1}{y} = \frac{1}{z} ....proved....$
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