# 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.

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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? (x 3 )(x 4 ) You add them!! (Inside—Add) (x 3 )(x 4 ) x (3 + 4) x7x7

DEF:  (1) Log b of a product log b (x · y) = log b x + log b y Inside parenthesis—Add logs

What do you with the exponents when they are in a quotient? x3x3 x4x4 You subtract them! (Quotient—Subtract) x3x3 x4x4 x (3 – 4)

DEF:  (2) Log b of a quotient log b (x / y) = log b x – log b y 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 3 7 + log 3 t – log 3 4 – log 3 v log 3 (7t 4v)

EX:  Write as one logarithm: log 2 9 + log 2 3c – log 2 c – log 2 5c log 2 (27c 5c 2 ) = 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? (x 3 ) 4 You multiply them!! (Outside—multiply) (x 3 ) 4 x34x34 x 12

DEF:  (3) Log b of a power Log b (x) r = r log b (x) Outside—Multiply (out in front 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 b 4  (m 3 / 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 (c 5 d 3 ) EX:4 1/3(log 5 t + 4 log 5 v – log 5 w) log 5 3  tv 4 /w

EX:  Expand log 2 (5c 2 / 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 (t 3  5 /v 4 ) (1/4)(log 2 3t + log 2 5v) log 2 ( 4  15tv)

Solve 2 x =  2 x = 2 -4 x = -4 64 x – 4 = (½) 2x (2 6 ) x – 4 = (2 -1 ) 2x 6x – 24 = -2x 8x = 24 x = 3

Solve (¾) 2x = 64 27 (¾) 2x = (27) -1 (64) (¾) 2x = (3 3 ) -1 (4 3 ) 2x = -3 x = -3 2 e 3x = e 7x – 2 3x = 7x – 2 -4x = -2 x = ½

Solve log n 1 = 2 25 n 2 = 1 25 n = 1 5 log √2 t = 6 (  2 ) 6 = t  64 = t or (2 1/2 ) 6 = t t = 8

Solve log 2 x 3 = 6 2 6 = x 3 64 = x 3 or (2 6 ) 1/3 = (x 3 ) 1/3 4 = x ln (3x – 5) = 0 log e (3x – 5) = 0 e 0 = 3x – 5 1 = 3x – 5 6 = 3x 2 = x

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