15.4 logs of products and quotients obj: to expand the logarithm of a product or quotient to...
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15.4 Logs of Products and Quotients
OBJ: To expand the logarithm of a product or quotient To simplify a sum or
difference of logarithms
What do you with the exponents when they are inside parenthesis?
(x3)(x4)
You add them!! (Inside—Add)
(x3)(x4)
x(3 + 4)
x7
DEF: (1) Logb of a product
logb(x · y) = logbx + logby
Inside parenthesis—Add logs
What do you with the exponents when they are in a quotient?
x3
x4
You subtract them! (Quotient—Subtract)
x3
x4
x(3 – 4)
DEF: (2) Logb of a quotient
logb(x / y) = logbx – logby
Quotient—Subtract logs
HW 5 p399 ( 2, 4, 6, 24, 26)P397 EX: 1 Expand log 3 (5c/d)
How many log 3 do I need to write?
log 3 log 3 log 3 log 3 ___
log 3 5 +
log 3 c –
log 3
– log 3 d
EX: Expand log b (7m/3n)
log b 7 +
log b m –
log b 3
–log b n
HW 5 P 399 (10, 12,14, 28)P398EX:3Write as one logarithm:
log 37 + log 3 t – log 34 – log 3v
log 3 (7t
4v)
EX: Write as one logarithm:
log 2 9 + log 2 3c – log 2 c – log 25c
log 2 (27c
5c2) =
log 2 (27
5c)
15.5 Logs of Powers and Radicals
OBJ: To expand a logarithm of a power or a radical
To simplify a multiple of a logarithm
What do you with the exponents when they are Outside parenthesis?
(x3)4
You multiply them!! (Outside—multiply)
(x3)4
x34
x12
DEF: (3) Logb of a power
Logb(x)r = r logb (x)
Outside—Multiply (out infront of log)
HW 5 P402 ( 2, 4, 6, 26, 28)P 400 EX1:Expand log 5 (mn 3 ) 2
How many log 5 do I write in the
parenthesis and with what symbol in
between?
(log 5 m + log 5 n)
Where does the outside exponent of
2 and the inside exponent of 3 go?
2 (log 5 m + 3log 5 n)
P400 EX : 1 Expand log b4(m3/ n)
What does 4 (read as fourth root)
become out in front of the parenthesis and
what symbol is separating the terms in the
parenthesis?( )( – )
(1/4)(3log b m – log b n)
HW5 P402 (10,12,14,32,34) P401EX:3 Write as one log:
5 log 2 c + 3 log 2 d
log 2 (c5d3)
EX:4 1/3(log5 t + 4 log5 v – log5 w)
log 5 3 tv4/w
EX: Expand
log 2 (5c2 / d ) 3
3(log 2 5 + 2log 2 c – log 2 d)
log 5 3 4x 2
(1/3)(log 5 4 + 2log 5 x)
EX: Write as one log:
3 log 2 t + ½ log 2 5 – 4 log 2 v
log 2 (t35 /v4)(1/4)(log 2 3t + log 2 5v)
log 2 (415tv)
Solve
2x = 2x = 2-4
x = -4
64x – 4 = (½)2x
(26) x – 4 = (2-1)2x
6x – 24 = -2x
8x = 24
x = 3
Solve
(¾)2x = 64
27
(¾)2x = (27)-1
(64)
(¾)2x = (33)-1
(43)
2x = -3
x = -3
2
e3x = e7x – 2
3x = 7x – 2
-4x = -2
x = ½
Solve
logn 1 = 2
25
n2 = 1
25
n = 1
5
log√2 t = 6
(2 )6 = t
64 = t or
(21/2)6 = t
t = 8
Solve
log2 x3 = 6
26 = x3
64 = x3 or
(26)1/3 = (x3)1/3
4 = x
ln (3x – 5) = 0
loge (3x – 5) = 0
e0 = 3x – 5
1 = 3x – 5
6 = 3x
2 = x