1. Exponents, collecting terms and log rules

2. Exponent rules:
xaxb=

3. Exponent rules:
xaxb=x(a+b) (xa)b=

4. Exponent rules:
xaxb=x(a+b) (xa)b=xab xa/xb=

5. Exponent rules:
xaxb=x(a+b) (xa)b=xab xa/xb=x(a-b) x0=

6. Exponent rules:
xaxb=x(a+b) (xa)b=xab xa/xb=x(a-b) x0=1

7. Try it:
2324= (52)3= 34/32= 70=

8. Try it:
2324=2(3+4) (52)3=52∙3 34/32=3(4-2) 70=1

9. Try it:
2324=2(3+4)=27 (52)3=52∙3=56 34/32=3(4-2)=32 70=1

10. Try it:
2324=2(3+4)=27=128 (52)3=52∙3=56= /32=3(4-2)=32=9 70=1

11. The most common error:
What is 33?

12. The most common error: What is 33? 33= 3 x 3 x 3… not 3+3+3 nor 3 x 3.
By definition.

13. Fractional exponents.
By definition;
(x1/2) (x1/2) =

14. Fractional exponents. By definition; (x1/2) (x1/2) = (x1/2+1/2) = x1=x
Therefore, (x1/2) = √x
Fractional exponents are roots.

15. Fractional exponents. Try it. 16 1/2 = 27 1/3 = 8 2/3 = 32 3/5 =
27 1/3 =
8 2/3 =
32 3/5 =
125 4/3 =

16. Fractional exponents. Try it. 16 1/2 = 4 27 1/3 = 3 8 2/3 = 4
27 1/3 = 3
8 2/3 = 4
32 3/5 = 8
125 4/3 = 625

17. Collecting terms: (xaybzc)(xdyfzg)= xa+3xa+y+yb=
Collect exponents to identical bases when multiplying
(xaybzc)(xdyfzg)=
Collect identical base to exponent terms when adding.
xa+3xa+y+yb=

18. Collecting terms: (xaybzc)(xdyfzg)=xa+dyb+fzc+g xa+3xa+y+yb=4xa+y+yb
Collect exponents to identical bases when multiplying
(xaybzc)(xdyfzg)=xa+dyb+fzc+g
Collect identical base to exponent terms when adding.
xa+3xa+y+yb=4xa+y+yb

19. Try it (315372)(347351)= xy2+x2+y+2xy2=
Collect exponents to identical bases when multiplying
(315372)(347351)=
Collect identical base to exponent terms when adding.
xy2+x2+y+2xy2=

20. Try it (315372)(347351)=355475 xy2+x2+y+2xy2=3xy2+x2+y
Collect exponents to identical bases when multiplying
(315372)(347351)=355475
Collect identical base to exponent terms when adding.
xy2+x2+y+2xy2=3xy2+x2+y

21. Try it (3x5372)(347y52)= x+x2+x+2x2=
Collect exponents to identical bases when multiplying
(3x5372)(347y52)=
Collect identical base to exponent terms when adding.
x+x2+x+2x2=

22. Try it (3x5372)(347y52)=3x+4557y+2 x+x2+x+2x2=2x+3x2
Collect exponents to identical bases when multiplying
(3x5372)(347y52)=3x+4557y+2
Collect identical base to exponent terms when adding.
x+x2+x+2x2=2x+3x2

23. It is used to find the exponent
Log rules.
The log function is a function that states an exponent.
If ab=c then
loga(c)=b
It is used to find the exponent

24. 3 5 =
125
log ( 25 )= 2
The exponent
The base

25. log and ln (that’s “ell-en”)
There are two log buttons on your calculator
and
They use different bases (10 and e), but they can both do the job.
To raise a number to an exponent, use the button.
log ln ^
If you need to check—look at the inverse (2nd) functions

26. To find a log of any other base (not 10 or e)
loga(b)=log10(b)/log10(a)
and
loga(b)=ln(b)/ln(a)

27. For example:
log4(64)=log10(64)/log10(4)
and
log5(625)=ln(625)/ln(5)

28. Rules of logs: Without a calculator!
log (ab) =
Based on the rules of exponents!

29. Rules of logs: Without a calculator!
log (ab) = log(a) + log(b)
log (a/b) =
Based on the rules of exponents!

30. Rules of logs: Without a calculator!
log (ab) = log(a) + log(b)
log (a/b) = log(a) - log(b)
log (ab) =
Based on the rules of exponents!

31. Rules of logs: Without a calculator!
log (ab) = log(a) + log(b)
log (a/b) = log(a) - log(b)
log (ab) = b log (a)
log (1/a) =
Based on the rules of exponents!

32. Rules of logs: Without a calculator!
log (ab) = log(a) + log(b)
log (a/b) = log(a) - log(b)
log (ab) = b log (a)
log (1/a) = - log (a)
Based on the rules of exponents!

33. Rules of logs: Without a calculator!
log (ab) = log(a) + log(b)
log (a/b) = log(a) - log(b)
log (ab) = b log (a)
log (1/a) = - log (a)
…and log 1 = 0 in any base.
Based on the rules of exponents!

34. Log equations Remember: the log is the exponent.
Do a little algebra, rewrite as an exponent, solve.

35. ln x = 3.2

36. 5(x+3) -4 =75

37. 3 + log4 x =24

39. Write as a single log expression:
log (3) +log (x)+2log(y)-4(log (z)+2log(x))

40. Write as a single log expression: