Question

Prove that :

Answer 1

$\bigstar \: {\large{\textsf{\textbf{\underline{\underline{Concept :}}}}}}$

• Some of the properties of the log are as follows :

1) Product law -

$\star \: \sf log(mnp) = \underline{{\boxed{{\sf logm +logn + logp}}}}$

2) Division law -

$\star \: \sf log( \dfrac{m}{n} ) = \underline{{\boxed{{\sf logm - logn }}}}$

3) Power law -

$\star \: \sf log \: {m}^{n} = \underline{{\boxed{{\sf n \: log m }}}}$

4) We should also know that -

$\star \: \sf log 10 = \underline{{\boxed{{\sf 1 }}}}$

• 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!

$\: {\large{\textsf{\textbf{\underline{\underline{Solution:}}}}}}$

$\sf \red{ Question\: 1}$

Taking L.H.S

$\sf log630$

✦ Prime factors of 630 -

$\begin{gathered}\begin{gathered}{\begin{array}{ c|c}2&630 \\ \hline 3&315 \\ \hline 3&105\\ \hline5&35 \\ \hline 7&7\\ \hline &1\end{array}}\end{gathered}\end{gathered}$

$\implies \sf log(2 \times {3}^{2} \times 5 \times 7)$

• Using first property

$\implies \sf log2 + \underline{log {3}^{2} }+ log5 + log 7$

• Using third property

$\implies \sf log2 +2 \: log3 + log5 + log 7$

$\therefore \: \sf log630 = \underline{{\boxed{ \red{{\sf log2+2 \: log3 + log5 + log7}}}}}$

$\sf \green{ Question\: 2}$

Taking L.H.S

$\sf log10 + log100 + log1000 + log10000$

$\implies \sf log10 + log {10}^{2} + log {10}^{3} + log {10}^{4}$

• Using third property

$\implies \sf log10 +2 \: log 10 + 3 \: log 10 + 4 \: log 10$

• As we know log10 = 1

$\implies \sf 1+2 (1) + 3 (1)+ 4 (1)$

$\implies \sf 1+2 + 3 + 4$

$\implies \sf \green{10}$

$\therefore \: \sf log10 + log100 + log1000 + log10000= \underline{{\boxed{ \green{{\sf 10}}}}}$

$\rule{280pt}{2pt}$

Answer 2

Number 1

log 630

=log (2×3×3×5×7)

=log2+log 3^2-log 5+log7

= log 2+2 log 3 + log 5 + log 7

Number 2

log 10+ log 100+ log 1000+ log 10000

= log 10+2 log 10+ 3 log 10+ 4 log 10

= 10 log 10

= 10