[tex] \bigstar \: {\large{\textsf{\textbf{\underline{\underline{Concept :}}}}}}[/tex]
• Some of the properties of the log are as follows :
1) Product law -
[tex] \star \: \sf log(mnp) = \underline{{\boxed{{\sf logm +logn + logp}}}}[/tex]
2) Division law -
[tex] \star \: \sf log( \dfrac{m}{n} ) = \underline{{\boxed{{\sf logm - logn }}}}[/tex]
3) Power law -
[tex] \star \: \sf log \: {m}^{n} = \underline{{\boxed{{\sf n \: log m }}}}[/tex]
4) We should also know that -
[tex] \star \: \sf log 10 = \underline{{\boxed{{\sf 1 }}}}[/tex]
• To prove R.H.S equals to L.H.S, there exists three conditions :
1) To make R.H.S equals to L.H.S
2) To make L.H.S equals to R.H.S
3) To simplify both side equations and make them equal to a single value
In proving these questions, we're going to apply 2nd condition though all the three conditions described above are convertible to each other.
Now, let's start!
[tex] \: {\large{\textsf{\textbf{\underline{\underline{Solution:}}}}}}[/tex]
[tex] \sf \red{ Question\: 1}[/tex]
Taking L.H.S
[tex] \sf log630[/tex]
✦ Prime factors of 630 -
[tex]\begin{gathered}\begin{gathered}{\begin{array}{ c|c}2&630 \\ \hline 3&315 \\ \hline 3&105\\ \hline5&35 \\ \hline 7&7\\ \hline &1\end{array}}\end{gathered}\end{gathered}[/tex]
[tex] \implies \sf log(2 \times {3}^{2} \times 5 \times 7)[/tex]
• Using first property
[tex] \implies \sf log2 + \underline{log {3}^{2} }+ log5 + log 7[/tex]
• Using third property
[tex] \implies \sf log2 +2 \: log3 + log5 + log 7[/tex]
[tex] \therefore \: \sf log630 = \underline{{\boxed{ \red{{\sf log2+2 \: log3 + log5 + log7}}}}}[/tex]
[tex]\sf \green{ Question\: 2}[/tex]
Taking L.H.S
[tex] \sf log10 + log100 + log1000 + log10000[/tex]
[tex] \implies \sf log10 + log {10}^{2} + log {10}^{3} + log {10}^{4} [/tex]
• Using third property
[tex] \implies \sf log10 +2 \: log 10 + 3 \: log 10 + 4 \: log 10[/tex]
• As we know log10 = 1
[tex]\implies \sf 1+2 (1) + 3 (1)+ 4 (1)[/tex]
[tex]\implies \sf 1+2 + 3 + 4 [/tex]
[tex]\implies \sf \green{10}[/tex]
[tex] \therefore \: \sf log10 + log100 + log1000 + log10000= \underline{{\boxed{ \green{{\sf 10}}}}}[/tex]
[tex]\rule{280pt}{2pt}[/tex]
