1. WARM - UP Evaluate: log 3 81 Solve for x: log5 (2x+3) = log5 (4x -3)
Graph: log 2 (𝑥+1) −3

2. Change of Base Formula, Expanding & Condensing
Logarithms

3. OBJECTIVES Students will be able to...
Use logarithmic Properties for condensing and expanding logarithmic expressions
Evaluate logs and natural logs
Use the change of base formula to evaluate logs with bases other than 10

4. HOMEWORK Worksheets “The Change of Base Formula”
“Properties of Logarithms” – Expanding and Condensing

5. Change of Base Formula
Let a, b and x be positive real numbers such that 𝑎≠1 and b≠1. Then log 𝑎 𝑥 can be converted to a different base as follows.
Base b Base Base e
log 𝑎 𝑥 = log 𝑏 𝑥 log 𝑎 𝑥 log 𝑎 𝑥 = log 𝑥 log 𝑎 log 𝑎 𝑥 = ln 𝑥 ln 𝑎

6. EXAMPLES With logarithms: 1) log 4 25 2) log 2 12 With natural logs:

7. practice “Change of Base Formula” Worksheet
___________ minutes; stopping at __________

8. Properties of logarithms
Let a be a positive number such that 𝑎≠1, and let n be a real number. If u and v are positive real numbers, the following properties are true:
Log w/ Base a Natural Log
Product Property:
Quotient Property:
Power Property:
log 𝑎 𝑢𝑣 = log 𝑎 𝑢 + log 𝑎 𝑣
ln 𝑢𝑣 = ln 𝑢 + ln 𝑣
log 𝑎 𝑢 𝑣 = log 𝑎 𝑢 − log 𝑎 𝑣
ln 𝑢 𝑣 = ln 𝑢 − ln 𝑣
log 𝑎 𝑢 𝑛 = 𝑛 log 𝑎 𝑢
ln 𝑢 𝑛 =𝑛 ln 𝑢

9. Examples Write each logarithm in terms of ln 2 and ln 3
1) ln ) ln 2 27

10. examples
Find the exact value of each expression without using a calculator
1) log ) ln 𝑒 6 − ln 𝑒 2

11. Rewriting logarithmic expressions
Expanding Logarithmic Functions
Examples:
1) log 4 5 𝑥 3 𝑦 ) ln 3𝑥−5 7

12. Rewriting logarithmic expressions
Condensing Logarithmic Functions
Examples:
1) log 𝑥 +3 log (𝑥+1) 2) 2 ln 𝑥+2 − ln 𝑥 3) [ log 2 𝑥 + log 2 𝑥+1 ]

13. practice “Properties of Logarithms” Worksheet
___________ minutes; stopping at __________

14. CLOSURE
On the note card provided: (Hand in before walking out the door!)
1) Evaluate the logarithm using the change of base formula: log
2) Rewrite and simplify: ln (5 𝑒 6 )
3) Expand the logarithm: log 10 𝑦 2
4) Condense the logarithms: log 5 8 − log 5 𝑡