2. Examples-1 (simplify)
1. Find `2log(x)+3log(y)`
Solution:
`2log(x)+3log(y)`
`=log(x^2)+log(y^3)`
`=log((x^2) xx (y^3))`
`=log(x^2y^3)`
2. Find `log(20)+log(30)-1/2log(36)`
Solution:
`log(20)+log(30)-1/2log(36)`
`=log(20)+log(30)-(1/2)log(6^(2))`
`=log(20)+log(30)-(1/2)*2log(6)`
`=log(20)+log(30)-log(6)`
`=log(20 xx 30)-log(6)`
`=log(600)-log(6)`
`=log(600/6)`
`=log(100)`
`=log(10^(2))`
`=2log(10)`
`=2*1`
`=2`
3. Find `log(9,27)-log(27,9)`
Solution:
`log(9,27)-log(27,9)`
`=((log(27))/(log(9)))-log_(27)9`
`=(log(27))/(log(9))-log_(27)9`
`=(log(3^(3)))/(log(9))-log_(27)9`
`=(3log(3))/(log(9))-log_(27)9`
`=(3log(3))/(log(3^(2)))-log_(27)9`
`=(3log(3))/(2log(3))-((log(9))/(log(27)))`
`=(3log(3))/(2log(3))-(log(9))/(log(27))`
`=(3log(3))/(2log(3))-(log(3^(2)))/(log(27))`
`=(3log(3))/(2log(3))-(2log(3))/(log(27))`
`=(3log(3))/(2log(3))-(2log(3))/(log(3^(3)))`
`=(3log(3))/(2log(3))-(2log(3))/(3log(3))`
`=3/2-2/3`
`=5/6`
4. Find `log(b,a)*log(c,b)*log(a,c)`
Solution:
`log(b,a)*log(c,b)*log(a,c)`
`=((log(a))/(log(b)))log_(c)(b)log_(a)(c)`
`=(log(a)log_(c)(b)log_(a)(c))/(log(b))`
`=(log(a)((log(b))/(log(c)))log_(a)(c))/(log(b))`
`=(log(a)log(b)log_(a)(c))/(log(b)log(c))`
`=(log(a)log(b)((log(c))/(log(a))))/(log(b)log(c))`
`=(log(a)log(b)log(c))/(log(b)log(c)log(a))`
`=1`
5. Find `log(a^2/(bc))+log(b^2/(ac))+log(c^2/(ab))`
Solution:
`log(a^2/(bc))+log(b^2/(ac))+log(c^2/(ab))`
`=log(a^2/(bc))+log(b^2/(ac))+log(c^2/(ab))`
`=log((a^2/(bc)) xx (b^2/(ac)))+log(c^2/(ab))`
`=log((ab)/(c^2))+log(c^2/(ab))`
`=log(((ab)/(c^2)) xx (c^2/(ab)))`
`=log(1)`
`=0`
6. Find `log(2,log(2,log(2,log(2,65536))))`
Solution:
`log(2,log(2,log(2,log(2,65536))))`
`log(2,log(2,log(2,65536)))=2``log(2,log(2,log(2,65536)))` `log(2,log(2,65536))=4``log(2,log(2,65536))` `log(2,65536)=16``log(2,65536)`
`=(log(65536))/(log(2))`
`=(log(2^(16)))/(log(2))`
`=(16log(2))/(log(2))`
`=16` `=log_(2)16` `=(log(16))/(log(2))` `=(log(2^(4)))/(log(2))` `=(4log(2))/(log(2))` `=4` `=log_(2)4` `=(log(4))/(log(2))` `=(log(2^(2)))/(log(2))` `=(2log(2))/(log(2))` `=2`
`=log_(2)2`
`=1`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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