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Mathematics

Solve for x :

log⁡ xlog⁡ 5=log⁡ 9log⁡ (13)\dfrac{\log \space x}{\log \space 5} = \dfrac{\log \space 9}{\log \space \Big(\dfrac{1}{3}\Big)}

Logarithms

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Answer

Given,

⇒log⁡ xlog⁡ 5=log⁡ 9log⁡ (13)⇒log⁡ xlog⁡ 5=log⁡ 32log⁡ 3−1⇒log⁡ xlog⁡ 5=2log⁡ 3−1log⁡ 3⇒log⁡ xlog⁡ 5=−2⇒log⁡ x=−2×log⁡ 5⇒log⁡ x=log⁡ 5−2⇒log⁡ x=log⁡ (125)⇒x=125.\Rightarrow \dfrac{\log \space x}{\log \space 5} = \dfrac{\log \space 9}{\log \space \Big(\dfrac{1}{3}\Big)} \\[1em] \Rightarrow \dfrac{\log \space x}{\log \space 5} = \dfrac{\log \space 3^2}{\log \space 3 ^{-1}} \\[1em] \Rightarrow \dfrac{\log \space x}{\log \space 5} = \dfrac{2\log \space 3}{-1\log \space 3} \\[1em] \Rightarrow \dfrac{\log \space x}{\log \space 5} = -2 \\[1em] \Rightarrow \log \space x = -2 \times \log \space 5 \\[1em] \Rightarrow \log \space x = \log \space 5 ^{-2} \\[1em] \Rightarrow \log \space x = \log \space \Big(\dfrac{1}{25}\Big) \\[1em] \Rightarrow x = \dfrac{1}{25}.

Hence, the value of x = 125\dfrac{1}{25}.

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