Unit – Exponential and Logarithmic Functions and Equations

Lesson: Properties of Logarithms (Text 3.3)

Warm-upSimplify the following:1. x2x3 2. 103104 3. e2e4

4. 5. 6.

7. (x2)3 8. (103)4 9. (e2)4

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).

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

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)

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)

Answers:

1.ln x + ln y + ln z2.log2 x + log2 y – log2 z3.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) – 3lnx4.ln((x-2)(x+2)) = ln(x2-4)5.log4 (x/(x+1)3)