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Pre-Calculus Properties of Logarithms and the Change of Base Formula


Properties of Logarithms 1. log1 0 log b 1 0 2. log10 1 3. log104

log b b 1 4

4.10log42 42 5. log Ma

6. log MN

M 7. log N

log b bp

blogb n n

a log M

log M log N

log M

log N

p


Examples. 1. log7x

7x 2. log y 7x 3. log 3y 4. log 7x2

5. log 7x

2


Examples. 1. log7x

log7

log x

7x 2. log y

log 7

log x

log y

7x 3. log 3y

log 7

log x

log 3

4. log7x2

5. log 7x

log7 2

log x2

2log7x

log y

log7

2log7

2log x

2log x


6. log7x2 y 7. ln5x

8. ln7e

9. ln7e2 x 3


6. log 7x2 y

log 7 log 7

7. ln5x

ln5

ln x

8. ln7e

ln7

ln e

9. ln7e2 x 3

1

log x2

log y 1/2

2log x

1 log y 2

ln7

ln7 ln e2 ln x 3 ln7 2ln e 3ln x 2 ln7 3ln x


Combine into a single logarithm. 10. 5log x log y 11. 3log7

2log x

12. 3log7

2log x

log y

13. 3log7

2log x

log y

Do you see the difference?


Combine into a single logarithm. 10. 5log x

log y

log x

5

11. 3log 7

2log x

log 73

12. 3log 7

2log x

log y

log 73 log y

13. 3log 7

2log x log 7

3

5

log y

log x y

log x2

73 log 2 x

log x2

73 y log 2 x

log y

log x

2

log y

Do you see the difference?

73 log 2 xy


Change of Base Formula log M log b M log b

log 3 12

log12 log3

or

ln12 ln3

I am not saying that log12 is the same as ln12, because its not. But the quotient between log 12 and log3 is the same as the quotient between ln12 and ln3.


Now you can graph equation such as y log 3 x Before, you were limited to logs with base 10 or e. Examples.

y

y

log 3 x

log x log3

log5 x 4

log x 4 log5


Graph y

log 3 x



Pre-Calculus Notes Properties of Logarithms and Change of Base Formula solutions