1. Using natural logarithms
Lesson 81
Using natural logarithms

2. Natural logarithms
When the base is e, the logarithm is called a natural logarithm
ex = a means ln a= x

3. Inverse properties of logarithms
eln x = x and ln ex = x
where x>0

4. Simplifying exponential and logarithmic expressions
Simplify elnp
elnp = p
Simplify ln e3c2+1=
= 3c2+1
simplfy eln2x
simplify ln e2d2+d

5. Properties of natural logarithms
Product property
ln ab = ln a + ln b
Quotient Property
ln (a/b) = ln a - ln b
Power Property
ln ap = p ln a

6. Applying properties of natural logarithms
Rewrite as a sum or difference
ln 6e= ln 6 + ln e
= ln 6 + 1
ln 3e = ln 3e- ln x=ln3 + ln e-ln x x
=ln lnx
ln e5x2 = 5x2 ln e = 5x2 (1) = 5x2

7. practice Write as a sum or difference- then simplify ln 10e ln 5e y
3 ln e6x2
ln( 4c3 )4 e2

8. Continuous exponential growth
The formula for exponential growth where interest is compounded continuously is
A = P ert

9. Evaluating logarithmic expressions
Lesson 87
Evaluating logarithmic expressions

10. Evaluating expressions of the form loga(bc)d
Use the properties of logs to evaluate
log4(16x)3 when x = 256
=3log4(16x)
=3(log416+log4x)
Since log416 = 2
= 3(2+log4x)
= 6 +3log4x
When x = 256
= 6+ 3log4 256= (4)= 18

11. evaluate ln(7e)2 = 2 ln(7e) =2 (ln 7+ln e) = 2(ln 7 + 1) = 2 ln 7 + 2
= 2(1.95) + 2 = 5.9

12. practice
Evaluate when r = 243
log3(27r)4
Evaluate
ln(12e)3

13. the change of base formula
remember the change of base formula where a is the new base

14. using the change of base formula
Convert log100(10x)2 to base 10. then evaluate when x = 1000
=2 log100 (10x)
=2( log 10x)
log 100
= 2 (log10 +logx)
log100
=2( 1 + log 1000)= 2( 1+3) = 4

15. use change of base formula
Convert to base e, then evaluate when x = 6
log4(2x)3= 3 log4(2x)
=3 (ln 2x)= 3 (ln 2 + ln x)
ln ln 4
= 3 ( ln2 + ln 6)
ln 4
= 3( ) =
1.3863

16. practice solve - change to base 10 log 100(1000x)3 when x = 10
change to base e
log9(3x)5 when x = 4

17. Solving log equations using the change of base formula
Solve x = 100
log = 9x
change to base 10
log = 9x
log 1000
2 = 9x 3
2 = 27x x = 2/27

18. practice
Solve for x:
32 4x = 8