1. EXPONENTS AND LOGARITHMS

2. e
e is a mathematical constant ≈ … Commonly used as a base in exponential and logarithmic functions: exponential function – ex natural logrithm – logex or lnx follows all the rules for exponents and logs
E is also limit of (1+1/n)n as n becomes large, is the sum of 1/i for i=0 to infinity

3. EXPONENTS
an where a is the base and x is the exponent an = a · a · a · … · a e3 = e * e * e a1 = a e1 = e a0 = 1 e0 = 1 a-n = 1 a n e-2 = 1 e 2

4. EXPONENTS
Using your calculator: 10x: base 10 ex: base e yx: base y Try: e2 = e1.5 =
FOR THOSE CALCULATORS that have y to the x

5. LAWS OF EXPONENTS
The following laws of exponents work for ANY exponential function with the same base

6. LAWS OF EXPONENTS
aman = am+n e3e4 = e3+4 = e7 exe4 = ex+4 a2e4 = a2e4 Try: e7e11 eyex

7. LAWS OF EXPONENTS
(am)n = amn (e4)2 = e4*2 = e8 (e3)3 = e3*3 = e9 (108)5 = 108*5 = 1040 Try: (e2)2 (104)2

8. LAWS OF EXPONENTS
(ab)n = anbn (2e)3 = 23e3 = 8e3 (ae)2 = a2e2 Try: (ex)5 2(3e)3 (7a)2

9. LAWS OF EXPONENTS
a b n = a n b n e 3 2 = e = e 2 9 Try: e 2 3 2a e 3

10. LAWS OF EXPONENTS
a m a n = a m−n ≡ 1 a n−m e 7 e 8 = e 7−8 = e −1 = 1 e Try: e 4 e 2 3e 2 e 2

11. LAWS OF EXPONENTS
Try these: x −3 y 4 a 3 b −2 a −5 b 7 5 a 3 b 3a b 2 3

12. LOGARITHMS
The logarithm function is the inverse of the exponential function. Or, to say it differently, the logarithm is another way to write an exponent. Y = logbx if and only if by = x So, the logarithm of a given number (x) is the number (y) the base (b) must be raised by to produce that given number (x)

13. LOGARITHMS Logarithms are undefined for negative numbers
Recall, y = logbx if and only if by = x
blogbx = x eloge2 = eln2 = 2 (definition )
logaa = 1 logee = lne = 1 (lne = 1 iff e1 = e)
loga1 = 0 loge1 = ln1 = 0 (ln1 = 0 iff e0 = 1)
Substitute v and u in above definitions

14. LOGARITHM
Using your calculator: LOG: this is log10 aka the common log LN: this is loge aka the natural log x < 1, lnx < 0; x > 1, lnx > 0 Try: ln 0 = ln = ln 1 = ln 10 =

15. LAWS OF LOGARITHMS
logb(xz) = logbx + logbz ln(1*2) = ln1 + ln2 = 0 + ln2 = ln2 ln(3*2) = ln3 + ln2 ln(3*3) = ln3 + ln3 = 2(ln3) Try: ln(3*5) = ln(2x) =
Try both ways: ln15 = and ln3 + ln5 = = 2.708…

16. LAWS OF LOGARITHM logb x z = logbx – logbz loge 2 3 = ln2 – ln3
Try:
ln =
ln x 7 =
Try both ways 2/7 = ln0.28… = and ln2 – ln7 = – = -1.25…

17. LAWS OF LOGARITHMS
logb(xr) = rlogbx for every real number r loge(23) = 3ln2 loge(32) = 2ln3 Try: loge42 = logex3 = ln3x =

18. ln and e Recall, ln is the inverse of e Try: x = 2 x = 0.009 x lnx
elnx 1
1.5
e = 1.5 3
e = 3

19. EXAMPLES OF LOGARITHMS
Try: w/o calculator lne5 rewrite in condensed form: 2lnx + lny +ln8 3ln5 – ln19 expand: ln10x3

20. RADICALS a is the radicand n is the index of the radical
is called a radical
a is the radicand
n is the index of the radical
is the radical sign
by convention and is called square root

21. LAWS OF RADICALS
Laws of radicals follow the laws of exponents: Try:

22. SCIENTIFIC NOTATION Numbers written in the form a x 10b
when b is positive – move decimal point b places for the right
when b is negative – move decimal point b places to the left
Reverse the procedure for number written in decimal form
Follows the laws of exponents

23. EXAMPLES OF SCIENTIFIC NOTATION
1,003, x , x x 10-5