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SOLVING (expand and condense)

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Presentation on theme: "SOLVING (expand and condense)"β€” Presentation transcript:

1 SOLVING (expand and condense)
3.3: Properties of Logs SOLVING (expand and condense)

2 3.3 Properties of Logarithms
Essential Questions: How can I use the properties of logarithms to expand, condense, and solve logarithmic equations?

3 1st Type of Logarithm: Common Log οƒ  π₯𝐨𝐠 𝒙 =π’š Examples:
If there is no β€œb” term, the log has base 10. Examples: log 45 = ____ log = ____ log ΒΎ = _____ 1.65 .507 -.125

4 2nd type of Logarithm: Natural Log οƒ  π₯𝐧 𝒙 =π’š ln 4 loge4 = ___________
ln implies a base of β€œe” ln 4 loge4 = ___________ Examples: Ln 5 = _____ Ln 3.144= _____ Ln 5/8 = _____ 1.61 1.45 -.47

5 Laws of Natural Logarithms
ln XY = ln X + ln Y Product: Quotient: Power: One-to-One: Identity: ln 𝑋 π‘Œ = ln X – ln Y ln XY = Y ln X If ln X= ln Y, then X = Y If ln ex = x

6 Expand the logarithmic expression:
a) ln 7 𝑦 3 4 π‘₯ 3 ln ln y – ln 4 – 3 ln x OR ln ln y – (ln ln x) b) ln 6 π‘₯ 2 𝑦 4 ln ln x – 4 ln y

7 c) ln 6 𝑏 (1/2)ln6 – (1/2)lnb d) ln 4w 3 π‘₯ 2 ln4 + lnw + (2/3)lnx

8 Condense to a single logarithm
a) 5 ln(x + 1) + ln x b) 6 ln(x – 4) + 3 ln x c) ln (3x + 5) – 4ln x – 6ln(x – 1) ln x(x + 1)5 ln x3(x – 4)6 ln 3π‘₯+5 π‘₯ 4 (π‘₯βˆ’1 ) 6

9 One-to-One and Identity equations
Solve for x. a) ln 3 π‘₯ 2 = ln 4 b) ln ex+1 = 4 3 π‘₯ 2 =4 x + 1 = 4 x = 3 x2 = 64 x = Β± 8

10 Homework 3.3 Properties of Logs Worksheet
β€œNo, not my dog. I do my homework on my computer…and the cat ate the mouse.


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