free for all

2. LOGARITHMS
exponential logarithmic
m
b A logb ( A) m
b>0
A>0
Jeff Bivin -- LZHS

3. exponential logarithmic
m
b A logb ( A) m
2
3 9 log3 (9) 2
53 125 log5 (125) 3
3 1 1
2 8
log2 8 3
1 5
2 32 log1 (32) 5
2
y
x 2 log2 ( x) y
Jeff Bivin -- LZHS

4. Evaluate
log5 (25) u
u
5 25
u 2
5 5
u 2
log5 (25) 2
Jeff Bivin -- LZHS

5. Evaluate
log3 (81) u
u
3 81
u 4
3 3
u 4
log3 (81) 4
Jeff Bivin -- LZHS

6. Evaluate 1
log2 32 u
u 1
2 32
u 1
2 25
2u 2 5
u 5
1
log2 32 5
Jeff Bivin -- LZHS

7. Try this!!
log7 (7) u
u
7 7
u 1
7 7
u 1
log7 (7) 1
Jeff Bivin -- LZHS

8. Try this!!
Solve for x
logx(32) = 5
x5 = 32
x5 = 25
x = 2
Jeff Bivin -- LZHS

9. Try this!!
loga (1)
? 0
a 1
loga (1) 0
Jeff Bivin -- LZHS

10. Try this!!
loga (a)
? 1
a a
loga (a) 1
Jeff Bivin -- LZHS

11. Special Logarithms
log10 a log a
loge a ln a

12. Properties of Logarithms
 Product Property
 Quotient Property
 Power Property
Jeff Bivin -- LZHS

13. Product Property
m n m n
a a a
multiplication addition
logb (m n) logb (m) logb (n)
multiplication addition
Jeff Bivin -- LZHS

14. Product Property
log2 (16 4) log2 (16) log2 (4)
4 2 4 2
log2 (2 2 ) log2 (2 ) log2 (2 )
6
log2 (2 ) 4 2
6 6
Jeff Bivin -- LZHS

15. Quotient Property
m
a m n
division
n
a subtraction
a
m
log ( ) b n logb (m) logb (n)
division subtraction
Jeff Bivin -- LZHS

16. Quotient Property
32
log2 4 log2 (32) log2 (4)
5 2
log2 8 log2 (2 ) log2 (2 )
3
log2 (2 ) 5 2
3 3
Jeff Bivin -- LZHS

17. Power Property
m n mn
a a
p logb(m p )
logb(mp ) = p•logb(m)
Jeff Bivin -- LZHS

18. Power Property
7
log2 2 7 log2 (2)
7 71
7 7
Jeff Bivin -- LZHS

19. Power Property
n
log a a n
n.log a ( a )
n.1
n
Jeff Bivin -- LZHS

20. Change of Base Formula
logb x
loga x
logb a
log x
log a x
log a
Jeff Bivin -- LZHS

21. log b 1 1
log a b
log a log a logb a
log b
log b log c log c
log a b.logb c . log a c
log a log b log a
n
n log b n.log b n
log am b m
.log a b
log a m.log a m
Jeff Bivin -- LZHS

22. Expand 3 2
log5 ( x y )
3 2
product property log5 ( x ) log5 ( y )
power property 3 log5 ( x) 2 log5 ( y)
Jeff Bivin -- LZHS

23. Expand x7
log 5 y2 z5
7 2 5
quotient property log5 ( x ) log5 ( y z )
product property log5 ( x )7 2
log5 ( y ) 5
log5 ( z )
distributive property log5 ( x )7 2
log5 ( y ) 5
log5 ( z )
power property 7 log5 ( x) 2 log5 ( y) 5 log5 ( z)
Jeff Bivin -- LZHS

24. Condense
5 log3 x 6 log3 y 2 log3 z
power property log3 x 5
log3 y 6
log3 z 2
product property log3 x y 5 6
log3 z 2
x5 y 6
quotient property log 3 z2
Jeff Bivin -- LZHS

25. Condense
1
2 log10 x 2 log10 y 4 log10 z
1
2 4
Power property log10 x 2
log10 y log10 z
1
2 4
group / factor log10 x 2
log10 y log10 z
1
product property
2 4
log10 x 2
log10 y z
1
quotient property log10 x2
log x
y2z4 10 y 2 z 4
Jeff Bivin -- LZHS

26. Logarithm Equations
Properties
log a B log a C B C
Jeff Bivin -- LZHS

27. Solve for x
log3 3x 9 log3 x 3
3x 9 x 3
2x 12
x 6
Jeff Bivin -- LZHS

28. Solve for x
log3 3x 9 log3 x 3
check x 6
log3 3(6) 9 log3 6 3
log3 18 9 log3 6 3
log3 9 log3 9
checks!
Jeff Bivin -- LZHS

29. Solve for x
log3 3x 9 log3 x 3
3x 9 x 3
2x 12
x 6
6
Jeff Bivin -- LZHS

30. Solve for x
log4 7 log4 n 2 log4 6n
log4 7(n 2) log4 6n
7n 14 6n
n 14
Jeff Bivin -- LZHS

31. Solve for x
log4 7 log4 n 2 log4 6n
check n 14
log4 7 log4 14 2 log4 6(14)
log4 7 log4 12 log4 84
log4 7(12) log4 84
log4 84 log4 84
checks!
Jeff Bivin -- LZHS

32. Solve for x
log4 7 log4 n 2 log4 6n
log4 7(n 2) log4 6n
7n 14 6n
n 14
14
Jeff Bivin -- LZHS

33. Solve for x
log2 x 1 log2 x 1 3
3
log 2 ( x 1)( x 1) log 2 2
3
( x 1)( x 1) 2
2
x 1 8
2
x 9
x 3
Jeff Bivin -- LZHS

34. Solve for x
log2 x 1 log2 x 1 3
check x 3 check x 3
log2 3 1 log2 3 1 3 log2 3 1 log2 3 1 3
log2 4 log2 2 3 log2 2 log2 4 3
2 1 3 fails
3 3 The argument
checks! must be positive
Jeff Bivin -- LZHS

35. Solve for x
log2 x 1 log2 x 1 3
log2 ( x 1)(x 1) 3
3
2 ( x 1)(x 1)
2
8 x 1
2
9 x
3 x
3
Jeff Bivin -- LZHS

36. x x 3
3 27 3 3 x 3
x x 2
5 49 5 7 ????????

37. Solve exponential equation with
logarithms
x x 2
5 49 5 7 ????????
log 5 x log 7 2 x.log 5 2.log 7
log 7
x 2.
log 5
x 2.log 5 7

38. Solve for x
4x 2
log 5 log 7
(4 x) log(5) (2) log(7)
4 log(5) 4 log(5)
2 log( 7 )
x 4 log(5 )
log(7 2 )
x log(54 )
x log 625 (49)
Jeff Bivin -- LZHS

39. Solve for x
x 2 x 1
log 3 log 5
( x 2) log(3) ( x 1) log(5)
x log(3) 2 log(3) x log(5) 1log(5)
x log(3) x log(5) log(5) 2 log(3)
x log(3) log(5) log(5) 2 log(3)
5
log( ) 5
x 9
3
x log 3 ( )
log( ) ( ) 9
5 5
Jeff Bivin -- LZHS

40. Try this
3x 1
5 7 2
5 3x 1
log 7 log 2
5
log(7 ) (3x 1) log(2)
5
log( 7 ) 3x log(2) 1 log(2)
5
log(7 ) log(2) 3x log(2)
log( 10 )
log(8)
7
x
10
log ( )
Jeff Bivin -- LZHS 8 7
x

41. Try this
3x 2
ln 15 ln e
ln(15) 1
(3x 2) ln(e)
ln(15) 3x 2
ln(15) 2 3x
ln(15) 2
3 x
Jeff Bivin -- LZHS