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