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Warm-up Simplify the following: 1. x 2 x 3 2. 10 3 10 4 3. e 2 e 4 4. 5. 6.

Unit – Exponential and Logarithmic Functions and Equations Lesson: Properties of Logarithms (Text 3.3). Warm-up Simplify the following: 1. x 2 x 3 2. 10 3 10 4 3. e 2 e 4 4. 5. 6. 7. (x 2 ) 3 8. (10 3 ) 4 9. (e 2 ) 4. Change of Base Formula.

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Warm-up Simplify the following: 1. x 2 x 3 2. 10 3 10 4 3. e 2 e 4 4. 5. 6.

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  1. Unit – Exponential and Logarithmic Functions and EquationsLesson: Properties of Logarithms (Text 3.3) Warm-up Simplify the following: 1. x2x3 2. 103104 3. e2e4 4. 5. 6. 7. (x2)3 8. (103)4 9. (e2)4

  2. Change of Base Formula • Even though your calculator only has base 10 and base e logs, you can calculate logs with other bases by using logb a = or = . • Examples: log2 8 = = 3 (Try it on your calculator.) Then try log3 81. You should get 4. With log5 100, you should get 2.861 (to 3 decimal places).

  3. Properties of Logs • loga (uv) = loga u + loga v • loga (u/v) = loga u - loga v • loga un = n(loga u) • Examples: • log3 (2x) = log3 2 + log3 x • log7 (x/y) = log7 x – log7 y • log4 x2 = 2(log4 x) • Works for natural logs too. • ln 3x = ln 3 + ln x • ln x3y = ln x3 + ln y = 3ln x + ln y

  4. Examples (cont.) • Let’s go backwards! • ln x – ln 2 = ln (x/2) • log3 x – 2log3 y = log3 x – log3 y2 = log3 (x/y2) • 3ln x + 2ln y – 4ln z = ln x3 + ln y2 – ln z4 = ln x3y2 – ln z4 = ln (x3y2/z4) • ln x – 2[ln(x+2) + ln(x-2)] = ln x -2ln(x2-4) = ln x – ln(x2-4)2 = ln(x/(x2-4)2)

  5. Practice Problems • Expand: 1. ln xyz 2. log2 (xy/z) 3. ln [(x2-1)/x3] • Condense: 4. ln (x-2) + ln (x+2) 5. log4 x – 3log4 (x+1)

  6. Answers: • ln x + ln y + ln z • log2 x + log2 y – log2 z • ln (x2-1) – ln x3 = ln((x-1)(x+1)) –ln x3 • = ln(x-1) + ln(x+1) – ln x3 = ln(x-1) + ln(x+1) – 3lnx • ln((x-2)(x+2)) = ln(x2-4) • log4 (x/(x+1)3)

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