Properties of Logarithms

Section 3.3

3.3 PROPERTIES OF LOGARITHMS

• Change of base formula:

loga x = logb x / logb a = log x / log a

= ln x / ln a

1. Evaluate log9 1043.

log10 1043 / log10 9log 1043 / log 93.1630

2. Evaluate using ln: log9 1043. ln 1043 / ln 93.1630 Same!

• Properties of Logarithms:

loga (uv) = loga u + loga v

ln (uv) = ln u + ln v

loga u/v = loga u – loga v

ln u/v = ln u – ln v

loga un = n loga u

ln un = n ln u

Write each log in terms of ln 2 & ln 3:

3. ln 6

ln (2 · 3)

ln 2 + ln 3

4. ln 2/27

ln 2 – ln 27

ln 2 – ln 33

ln 2 – 3 ln 3

5. Verify that – log10 1/100 = log10 100

-log10 100-1 =

-(-1) log10 100 =

log10 100 =

Expand each expression:

6. log4 5x3y

log4 5 + log4 x3 + log4 y

log4 5 + 3 log4 x + log4 y

7. ln √3x – 5

7

ln (3x-5)1/2

7

ln (3x – 5)1/2 – ln 7

½ ln (3x – 5) – ln 7

Condense each expression:

8. ½ log10 x + 3 log10 (x + 1)

log x1/2 + log (x + 1)3

log [ √ x (x + 1)3]

9. 2 ln (x + 2) – ln x

ln (x + 2)2 – ln x

ln (x + 2)2

x

10.1/3 [log2 x + log2 (x – 4)]

1/3 {log2 [x(x – 4)]}

log2 [x(x – 4)]1/3

log2 3√ x(x – 4)

Homework

• Page 211-213

10-20 even, 35, 37-55 odd, 72, 74