C log a? Definition Notes

Transcription

1 Log Page C 8. log a? Definition Notes The Definition of a Logarithm: log 9? log 9? Think: What power do you have to raise 3 to, to equal 9? log 8? log 83? 3 8 equals to what power? b is the base. a is the thing you are logging c is the answer/exponent Switching from Log Form to : Log Form The thing you are Logging equals the Base to the other side. Remember: The base of the log is the base of the exponent. The exponent is the Answer. Log Form log 6? log 6 4? 4 6 equals to what power? 6?? 4 log x x Exponent Laws x 4 Log Form > and for x log 6 x 6 x 4 log 6 4 Set Log arbitrarily x Same Base: Make exponents equal to each other log x x 3 log 6a x 6a a a a x 4 Exponent Laws

2 Log Page C 8. log xc, log ac,log ax Notes Log Form > and for x log 5 x x 3 Same Base: Make exponents equal to each other The base of the log is the base of the exponent log x3 x 4 x 64 log x x 6 x 36 log x x 5 x 5 Exponent Laws log x x 9 x 9 x 5 x 3 log x 4 x x 4 log x x x x 5 Fifth Root Both Sides log x 7 x 7 x 7 x x 9 Take both/sides to reciprocal exponent log x5 x 5 x 4 5 x 9 log x 3 x 3 x 5 log 6 x 6 4 x x OR log 6 x log 6 x 4 x x Do Algebra First! log 6 x 6 x 4 log 6 4

3 Log Page 3 C 8. Log Restrictions Notes State Restrictions logx x0 log 0 und log3 und log # x0,x log # und log # und log # und State Restrictions and Domain: Set the thing you are logging to greater than or equal to zero, then solve. log x x x 4 x0 log x5 x 5 x 4 5 x 9 x50 x 5 log 3x 3 3 x 3 x 8 x 5 3x0 x 3 x 3 log x x 3 x 9 x 9 x 3 x 0 x0,x0 x0 log x log x x x 9 x 9 x 3 x0 x 3, x 3 x 3, x 3 log 5x x 5x x 36 5x x 6 x 5x 6 0 x x 3 0 5x x 0 x5 x 0 0 x 5 log x x 9 x 3 x 4 0 x x 0 x 0 x x 0 x, x x, x 3 x, x log x 3 x 3x 3 x 6x9 0 x 6x7 0 x 7x x 7, x x30 x3 x 3 x 4 Set the base of the log 0 and and solve.

4 Log Page 4 C 8.3 log a mloganotes Bring Exponent down in front and Vice Versa log5 log5 logx logx 3log4 3log4 6log4 OR 3log4 log4 log4 log x logx log x logx log log log log log 5 x 5 5 x 4 5 to what power is 5 log 65 x log 5 x 4log 5x 4 x x 4 Bring Exponent down in front Log Rules log 5 log 65 x x 4

5 Log Page 5 C 8.3 Notes log6 log log log 6 log 4 log log 4 log 4 log 4 Choose the Base you want! Think about it. log 7 log7 log3 log3 log3 3log3 3 log3 Exponent down in front 7 3 log 6 log 6 4 log log log log8 log8 log log8 log log 83 8

6 Log Page 6 C 8.4 log mlog nlog mn log mlog nlog log 4 log 8log 48log Add Multiply 3 log 3 log 9log 39log Add Multiply 7 3 log log5 log7 log 5 7 log 35 Add Multiply 7 log 7 log 3log log Subtract Divide log4 log0 log0 log 40 log8 0 Positives on top, Negatives on Bottom 5 log5 log log0 log log 0 4 log5 log log0 log log5 50 log64 log4 6 log 4 log6 Separate into an addition of logs log5 log 0 log0 log Separate into a subtraction of logs

7 Log Page 7 C 8.4 log mlog nlog mn log mlog nlog log 3 logx log3x log 3x3 Add Multiply logx logx log xx log x x Add Multiply logxlogxlogxxlogx Add Multiply log x logx log x x logx Subtract Divide Simplify logx logx log x xx log logx x x Subtract Divide Factor Simplify

8 Log Page 8 C 8.5 Log Operation Notes log log Calculator Math, Alpha, Math log6 log Bring your exponent down in front. log5 3log5 6log If there is a number in front multiply log log log If 5 5 for example you know: log5 a log5 a log5 x log5 xlog5 log5 Distribute 3xlog7 xlog x3log7 log GCF x logxy logx logy logx logy The exponent only applies to the y value logx y logx logy logx logy logxy logxy logx logy logx logy You can bring this exponent down in front. Remember: If you separate into an addition you must distribute the. logx logxlogx Cannot bring exponent down in front log5 log log5 log 0.04 Cannot multiply logs logx logx logx logx logx logx logx logx Cannot Divide logs log5 log5 Cannot distribute into a log logx log x Cannot distribute

9 Log Page 9 C 8.5 Operation log a Notes log 8log 8 log 64 3 Take the base and the log to any exponent you like! log 4 log 4 log 4 log 4 Bring your exponent down in front log Take the base and the thing you are logging to an exponent to get like bases to use log laws

10 Log Page 0 C 8.6 Log/Delog Both Sides Notes 4 log4 log log4 xlog log4 x log log 4x 4 x Log Both Sides Bring Exponents Down In Front Divide Change of base Change of base 3 5 log3 log5 log3 xlog5 log3 x log5 log 3x x Algebraic answer Check Answer: 5. 3 Before you log both sides! 3 4 Add/Subtract First 8 4 Divide First Or 8 log8 log log8 log log 4 7 log4 log7 log4 x log7 log4 xlog7 log 7 log4 log7 xlog7 log4 log7 x log7 log4 log7 x log7 x 0.9 Distribute Or divide log7 and minus Combine x's on one side Everything else on other side Factor out x Divide 8 3 log8 log3 log8 xlog3 log8 x log3 log 8 x log 8x x log 8 Bring Fraction In Front. Bring Coefficient Up 9 log log9 x 5log x log9 xlog 5log xlog9 log9 xlog xlog9 log9 5log xlog log9 log9 5log log9 5log x log log log6 3 log4 log6 log3 log4 log6 xlog3 x 5log4 log6 xlog3 xlog4 5log4 xlog4 xlog3 log6 5log4 xlog4 log3 log6 5log4 log6 log5 x log4 log3 GCF x Divide De log Both sides log 4log x 4 x Rule 6 Proof a x log a logx log xloga logx log xloga logx loga log a log x logx loga log xlog x x x Remember: You may only log both sides if SAMD is complete. Bedmas backwards. Remember: If you do log a product you must separate into an addition of logs. Remember: if you log a sum you must use brackets Remember: You may only de log both sides if one log equals one log.

11 C 8.7 Graph Log Notes Graph: ylog x y x y x y 0 0 y xy ylog x VA: x0 Domain: x0 Graph: ylog x y y Left Up ylog x ylog x xy x0 x x0 x Log Page

12 Log Page C 8.7 Inverse Log Graphs Notes y ylog x,,4, 0, x y 0 4 x y 0 und 0 4,,0, 4, y x logx log logx ylog logx y log log xy y log x f x log x Switch x and y Log Both Sides Bring Exponents Down In Front Divide Change of base Mirror Inverse Function notation y x y log x f x log x Switch x and y Exponential to log Form Back the Other Way! y log x x log y y y f x y 3 x 3 x 3 logx 3 y log logx 3 y log log x3 y log x3y y log x 3 f x log x 3 Inverse Proof y log x3 x log y3 x log y 3 y3 3y y 3 f x 3 Remember: Inverse: Switch x and y Remember: A diagonal reflection over the line y x