![\bigstar \: {\large{\textsf{\textbf{\underline{\underline{Concept :}}}}}}](https://tex.z-dn.net/?f=%20%5Cbigstar%20%5C%3A%20%7B%5Clarge%7B%5Ctextsf%7B%5Ctextbf%7B%5Cunderline%7B%5Cunderline%7BConcept%20%3A%7D%7D%7D%7D%7D%7D)
• Some of the properties of the log are as follows :
1) <u>Product law </u><u>-</u>
![\star \: \sf log(mnp) = \underline{{\boxed{{\sf logm +logn + logp}}}}](https://tex.z-dn.net/?f=%20%5Cstar%20%5C%3A%20%5Csf%20log%28mnp%29%20%3D%20%5Cunderline%7B%7B%5Cboxed%7B%7B%5Csf%20logm%20%2Blogn%20%2B%20logp%7D%7D%7D%7D)
2) <u>Divisio</u><u>n</u><u> </u><u>law</u><u> </u><u>-</u>
![\star \: \sf log( \dfrac{m}{n} ) = \underline{{\boxed{{\sf logm - logn }}}}](https://tex.z-dn.net/?f=%20%5Cstar%20%5C%3A%20%5Csf%20log%28%20%5Cdfrac%7Bm%7D%7Bn%7D%20%29%20%3D%20%5Cunderline%7B%7B%5Cboxed%7B%7B%5Csf%20logm%20%20-%20logn%20%7D%7D%7D%7D)
3) <u>Power law -</u>
![\star \: \sf log \: {m}^{n} = \underline{{\boxed{{\sf n \: log m }}}}](https://tex.z-dn.net/?f=%20%5Cstar%20%5C%3A%20%5Csf%20log%20%5C%3A%20%20%7Bm%7D%5E%7Bn%7D%20%3D%20%5Cunderline%7B%7B%5Cboxed%7B%7B%5Csf%20n%20%5C%3A%20log%20m%20%7D%7D%7D%7D)
4) We should also know that -
![\star \: \sf log 10 = \underline{{\boxed{{\sf 1 }}}}](https://tex.z-dn.net/?f=%20%5Cstar%20%5C%3A%20%5Csf%20log%2010%20%3D%20%5Cunderline%7B%7B%5Cboxed%7B%7B%5Csf%201%20%7D%7D%7D%7D)
• 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.
<u>Now, let's </u><u>start</u><u>!</u>
![\: {\large{\textsf{\textbf{\underline{\underline{Solution:}}}}}}](https://tex.z-dn.net/?f=%20%5C%3A%20%7B%5Clarge%7B%5Ctextsf%7B%5Ctextbf%7B%5Cunderline%7B%5Cunderline%7BSolution%3A%7D%7D%7D%7D%7D%7D)
![\sf \red{ Question\: 1}](https://tex.z-dn.net/?f=%20%5Csf%20%5Cred%7B%20Question%5C%3A%201%7D)
<u>Taking</u><u> </u><u>L.H.S</u>
![\sf log630](https://tex.z-dn.net/?f=%20%5Csf%20log630)
✦ Prime factors of 630 -
![\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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5Cbegin%7Bgathered%7D%7B%5Cbegin%7Barray%7D%7B%20c%7Cc%7D2%26630%20%5C%5C%20%5Chline%203%26315%20%5C%5C%20%5Chline%203%26105%5C%5C%20%5Chline5%2635%20%5C%5C%20%5Chline%207%267%5C%5C%20%20%5Chline%20%261%5Cend%7Barray%7D%7D%5Cend%7Bgathered%7D%5Cend%7Bgathered%7D)
![\implies \sf log(2 \times {3}^{2} \times 5 \times 7)](https://tex.z-dn.net/?f=%20%5Cimplies%20%5Csf%20log%282%20%5Ctimes%20%20%7B3%7D%5E%7B2%7D%20%20%5Ctimes%205%20%5Ctimes%207%29)
• Using first property
![\implies \sf log2 + \underline{log {3}^{2} }+ log5 + log 7](https://tex.z-dn.net/?f=%20%5Cimplies%20%5Csf%20log2%20%20%2B%20%5Cunderline%7Blog%20%20%7B3%7D%5E%7B2%7D%20%7D%2B%20%20%20log5%20%20%2B%20log%207)
• Using <u>third</u> property
![\implies \sf log2 +2 \: log3 + log5 + log 7](https://tex.z-dn.net/?f=%20%5Cimplies%20%5Csf%20log2%20%20%2B2%20%5C%3A%20log3%20%2B%20%20%20log5%20%20%2B%20log%207)
![\therefore \: \sf log630 = \underline{{\boxed{ \red{{\sf log2+2 \: log3 + log5 + log7}}}}}](https://tex.z-dn.net/?f=%20%5Ctherefore%20%5C%3A%20%5Csf%20log630%20%3D%20%20%5Cunderline%7B%7B%5Cboxed%7B%20%5Cred%7B%7B%5Csf%20log2%2B2%20%5C%3A%20log3%20%2B%20log5%20%2B%20log7%7D%7D%7D%7D%7D)
![\sf \green{ Question\: 2}](https://tex.z-dn.net/?f=%5Csf%20%5Cgreen%7B%20Question%5C%3A%202%7D)
<u>Taking L.H.S</u>
![\sf log10 + log100 + log1000 + log10000](https://tex.z-dn.net/?f=%20%5Csf%20log10%20%2B%20log100%20%2B%20log1000%20%2B%20log10000)
![\implies \sf log10 + log {10}^{2} + log {10}^{3} + log {10}^{4}](https://tex.z-dn.net/?f=%20%5Cimplies%20%5Csf%20log10%20%2B%20log%20%7B10%7D%5E%7B2%7D%20%20%2B%20log%20%7B10%7D%5E%7B3%7D%20%2B%20log%20%7B10%7D%5E%7B4%7D%20)
• Using third property
![\implies \sf log10 +2 \: log 10 + 3 \: log 10 + 4 \: log 10](https://tex.z-dn.net/?f=%20%5Cimplies%20%5Csf%20log10%20%2B2%20%5C%3A%20%20log%2010%20%20%2B%203%20%5C%3A%20log%2010%20%2B%204%20%5C%3A%20log%2010)
• As we know log10 = 1
![\implies \sf 1+2 (1) + 3 (1)+ 4 (1)](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%201%2B2%20%20%281%29%20%2B%203%20%281%29%2B%204%20%281%29)
![\implies \sf 1+2 + 3 + 4](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%201%2B2%20%20%2B%203%20%2B%204%20)
![\implies \sf \green{10}](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%20%20%5Cgreen%7B10%7D)
![\therefore \: \sf log10 + log100 + log1000 + log10000= \underline{{\boxed{ \green{{\sf 10}}}}}](https://tex.z-dn.net/?f=%20%5Ctherefore%20%5C%3A%20%5Csf%20%20%20log10%20%2B%20log100%20%2B%20log1000%20%2B%20log10000%3D%20%20%5Cunderline%7B%7B%5Cboxed%7B%20%5Cgreen%7B%7B%5Csf%2010%7D%7D%7D%7D%7D)
![\rule{280pt}{2pt}](https://tex.z-dn.net/?f=%5Crule%7B280pt%7D%7B2pt%7D)