Question

if 2^a=5^b=10^c prove that 1/a+1/b=1/c​

Answer 1

Answer:-

$\sf{Let\:2^a = 5^b = 10^c = k}$

$\sf{\\}$

So,

$\sf{2^a = k}$

$\texttt{To the power 1/a both sides}$

$\longrightarrow \sf{2 = {k}^{\frac{1}{a}} \dots (i) }$

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And,

$\sf{5^b = k}$

$\texttt{To the power 1/b both sides}$

$\longrightarrow \sf{5 = {k}^{\frac{1}{b}} \dots (ii)}$

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Similarly,

$\sf{10^c = k}$

$\texttt{To the power 1/c both sides}$

$\longrightarrow \sf{10 = {k}^{\frac{1}{c}} \dots (iii) }$

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We know that,

$\sf{2 * 5 = 10}$

$\longrightarrow \sf{ {k}^{\frac{1}{a}} * {k}^{\frac{1}{b}} = {k}^{\frac{1}{c}} \quad \quad [From\:(i),\:(ii),\:and\:(iii)] }$

$\longrightarrow \sf{ {k}^{\frac{1}{a} + \frac{1}{b}} = {k}^{\frac{1}{c}} }$

As the bases are same,

$\longrightarrow \underline{ \underline{ \green{\sf{ \dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{c} }}}}$