Question
Prove that : $\log_{10} 125 = 3(1 - \log_{10}2).$
✓
Answer
$ \log _{10} 125=3\left(1-\log _{10} 2\right)$
$\text { L.H.S. }=\log _{10} 125$
$\Rightarrow \log _{10} 5 \times 5 \times 5$
$\Rightarrow \log _{10} 5^3$
$\Rightarrow 3 \log _{10} 5 \ldots .(1)$
$\text { R.H.S }=3\left(1-\log _{10} 2\right)$
$\left.\Rightarrow 3 \log _{10} 10-\log _{10} 2\right)$
$\Rightarrow 3 \log _{10}\left(\frac{10}{2}\right)$
$\Rightarrow 3 \log _{10} 5$
From $(1)$ and $(2),$ we have
$\text{L.H.S.} =\text {R. H. S}.$
Hence proved.
$\text { L.H.S. }=\log _{10} 125$
$\Rightarrow \log _{10} 5 \times 5 \times 5$
$\Rightarrow \log _{10} 5^3$
$\Rightarrow 3 \log _{10} 5 \ldots .(1)$
$\text { R.H.S }=3\left(1-\log _{10} 2\right)$
$\left.\Rightarrow 3 \log _{10} 10-\log _{10} 2\right)$
$\Rightarrow 3 \log _{10}\left(\frac{10}{2}\right)$
$\Rightarrow 3 \log _{10} 5$
From $(1)$ and $(2),$ we have
$\text{L.H.S.} =\text {R. H. S}.$
Hence proved.
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