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1. EXPONENTS AND LOGARITHMS

2. e e is a mathematical constant ≈ 2.71828… 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

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 = e-2 =

4. EXPONENTS Using your calculator: 10x: base 10 ex: base e yx: base y Try: e2 = e1.5 =

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 Try these:

10. 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)

11. 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)

12. 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 0.000001 = ln 1 = ln 10 =

13. 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) =

14. LAWS OF LOGARITHM logb= logbx– logbz loge = ln2 – ln3 loge = ln3 – ln5 Try: ln = ln =

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

16. ln and e Recall, ln is the inverse of e Try: x = 2 x = 0.009

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

18. RADICALS 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