Question

Prove that :

Q1) log 630 = log 2 + 2log 3 + log 5 + log 7
Q2) log 10+ log 100+ log 1000+ log 10000 = 10​

Answer 1

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

Answer 2

`\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( (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 -

`\implies \sf log(2 * {3}^(2) * 5 * 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}`