5.3 Properties of Logarithmic Functions and their Graphs

A. Expanding and Condensing Logarithms

B. Evaluate Logs without a Calculator

A.Properties: There are 3 big ones!

• 1. PRODUCT PROPERTY:• Loga(uv) = logau + logav• I can use this to expand something:• ln(xy) = • Or I can use this to condense something:• ln a + ln b =• WATCH OUT! You cannot do anything

with ln(a + b). So tempting.

2. Quotient Property:

• Loga(u/v) = logau – logav

• With this, I can expand things:

• ln(a/b) =

• Or I can condense things:

• ln a – ln b =

• But the temptations are lurking: We cannot do anything with ln (a – b).

3. Power Property

• Logaun = nlogau

• I can use this to expand expressions:

• lnx2 =

• Or I can use this to condense expressions:

• 4lny =

Now let’s do some practice with these 3 big properties.

Expand log45x3y

Condense (1/2)logx + 3log(x + 1)

7

53ln Expand

x

Condense 2ln(x + 2) - lnx

1loglog3

1 Condense 22 xx

2ln Expand

1

6ln Expand :Try

2

2

xx

x

1ln1lnln32 Condense

)4ln(58ln2 Condense :Try

xxx

z

B. Evaluate Logs

26

35

lnln Evaluate

5log Evaluate

ee

You try: (These are from the HW)

e

e

1ln

ln

8log

5.4

42

Try these (from the HW)

• Ln e2 + ln e5

• Log 575 – log 53