Compare-it-log-4-20-2-and-log-4-320-

Question Number 158814 by HongKing last updated on 09/Nov/21

Compare it :

{(\log_{4} 20)}^{2} and \log_{4} 320

Answered by Rasheed.Sindhi last updated on 09/Nov/21

{(\log_{4} 20)}^{2} \overset{?}{=} \log_{4} 320

{(\log_{4} (2^{2} . 5))}^{2} \overset{?}{=} \log_{4} (2^{6} . 5)

{({2 \log}_{4} 2 + \log_{4} 5)}^{2} \overset{?}{=} {6 \log}_{4} 2 + \log_{4} 5

{(2 (\frac{1}{2}) + \log_{4} 5)}^{2} \overset{?}{=} 6 (\frac{1}{2}) + \log_{4} 5

{(1 + \log_{4} 5)}^{2} \overset{?}{=} 3 + \log_{4} 5

1 + {(\log_{4} 5)}^{2} + {2 \log}_{4} 5 \overset{?}{=} 3 + \log_{4} 5

{(\log_{4} 5)}^{2} + \log_{4} 5 \overset{?}{=} 2

O b v i o u s l y \log_{4} 5 > \log_{4} 4 = 1

{(\log_{4} 5)}^{2} + \log_{4} 5 > {(\log_{4} 4)}^{2} + \log_{4} 4

{(\log_{4} 5)}^{2} + \log_{4} 5 > 1 + 1 = 2

∴ {(\log_{4} 20)}^{2} > \log_{4} 320

Commented by HongKing last updated on 09/Nov/21

thank you very much my dear Ser cool

Answered by Raxreedoroid last updated on 09/Nov/21

\log_{4} 20 = 1 + \log_{4} 5

{(\log_{4} 20)}^{2} = \log_{4} 20 + (\log_{4} 20) (\log_{4} 5)

\log_{4} 320 = \log_{4} 20 + \log_{4} 16

∴ {(\log_{4} 20)}^{2} > \log_{4} 320

Commented by HongKing last updated on 09/Nov/21

thank you very much dear Ser cool