Question

If (2.4)x= (0.24)y = 102 the show that 1/x - 1/z=1/y​

Answer 1

$Given \: (2.4)^{x} = (0.24)^{y} = 10^{z}$

$Let \: (2.4)^{x} = (0.24)^{y} = 10^{z} = k$

$i) (2.4)^{x} = k \implies 2.4 = k^{\frac{1}{x}} \: --(1)$

$ii) (0.24)^{y} = k \implies 0.24 = k^{\frac{1}{</p><p>y}} \: --(2)$

$iii) (10)^{z} = k \implies 10 = k^{\frac{1}{z}} \: --(3)$

/* Do (1) ÷ (3) , we get */

$\implies \frac{2.4}{10} = \frac{k^{\frac{1}{x}}}{k^{\frac{1}{z}}}$

$\implies 0.24 = k^{\frac{1}{x} - \frac{1}{z}}$

$\boxed{ \pink{ \because \frac{x^{m}}{x^{n}} = x^{m-n} }}$

$\implies k^{\frac{1}{y}} = k^{\frac{1}{x} - \frac{1}{z}} \: \blue {[ From \: (2)] }$

$\implies \frac{1}{y} = \frac{1}{x} - \frac{1}{z}$

$\boxed{ \blue{ \because If \:a^m = a^n \implies m = n }}$

$Hence, Proved$

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