Question

What is expand log 19800

Answer 1

When the instructions say to "expand", they mean that they've given me one log expression with lots of stuff inside it, and they want me to use the log rules to take the log apart into many separate log terms, each with only one thing inside its particular log. That is, they've given me one log with a complicated argument, and they want me to convert this to many logs, each with a simple argument.

in this case log 19800 can be written as log(198×100)

=log198+log100

=log(11×9×2)+log100

log11+2log3+log2+2log10

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Answer 2

Solution :-

First prime factorise 19800

$\begin{array}{r | l} 2 & 19800 \\ \cline{1-2} 2 & 9900 \\ \cline{1-2} 2 & 4950 \\ \cline{1-2} 3 & 2475 \\ \cline{1-2} 3 & 825 \\ \cline{1-2} 5 & 275 \\ \cline{1-2} 5 & 55 \\ \cline{1-2} 11 & 11 \\ \cline{1-2} & 1 \end{array}$

19800 = 2³ * 3² * 5² * 11

$\\ \\ \sf log19801 \\ \\ \\ \sf = log( {2}^{3} \times {3}^{2} \times {5}^{2} \times 11) \\ \\ \\ \sf = log {2}^{3} + log {3}^{2} + log {5}^{2} + log11 \\ \\ \\ \boxed{ \bf \because log(ab)= loga + logb} \\ \\ \\ \sf = 3log2 + 2log3 + 2log5 + log11 \\ \\ \\ \boxed{ \bf \because loga^m = m.loga}$