WARM - UP
Evaluate: log 3 81
Solve for x: log5 (2x+3) = log5 (4x -3)
Graph: log 2 (𝑥 + 1) − 3
CHANGE OF BASE FORMULA,
EXPANDING & CONDENSING
Logarithms
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
HOMEWORK
Worksheets
“The Change of Base Formula”
“Properties of Logarithms” – Expanding and Condensing
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
log 𝑎 𝑥 =
log𝑏 𝑥
log𝑎 𝑥
Base 10
log 𝑎 𝑥 =
Base e
log 𝑥
log 𝑎
log 𝑎 𝑥 =
ln 𝑥
ln 𝑎
EXAMPLES
With logarithms:
1) log 4 25
2) log 2 12
With natural logs:
3) log 4 25
4) log 2 12
PRACTICE
“Change of Base Formula” Worksheet
 ___________ minutes; stopping at __________
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
1) Product Property:
2) Quotient Property:
3) Power Property:
Natural Log
log 𝑎 𝑢𝑣 = log 𝑎 𝑢 + log 𝑎 𝑣
ln 𝑢𝑣 = ln 𝑢 + ln 𝑣
𝑢
= log 𝑎 𝑢 − log 𝑎 𝑣
𝑣
𝑢
ln
= ln 𝑢 − ln 𝑣
𝑣
log 𝑎
log 𝑎 𝑢𝑛 = 𝑛log 𝑎 𝑢
ln 𝑢𝑛 = 𝑛 ln 𝑢
EXAMPLES
Write each logarithm in terms of ln 2 and ln 3
1) ln 6
2) ln
2
27
EXAMPLES
Find the exact value of each expression without using a calculator
3
1) log 5 5
2) ln 𝑒 6 − ln 𝑒 2
REWRITING LOGARITHMIC EXPRESSIONS
Expanding Logarithmic Functions
Examples:
1) log 4
5𝑥 3 𝑦
2) ln
3𝑥−5
7
REWRITING LOGARITHMIC EXPRESSIONS
Condensing Logarithmic Functions
Examples:
1)
1
log 𝑥
2
+ 3 log(𝑥 + 1)
2) 2 ln 𝑥 + 2 − ln 𝑥
3)
1
[log 2 𝑥
3
+ log 2 𝑥 + 1 ]
PRACTICE
“Properties of Logarithms” Worksheet
 ___________ minutes; stopping at __________
CLOSURE
On the note card provided: (Hand in before walking out the door!)
1) Evaluate the logarithm using the change of base formula:
log15 1250
2) Rewrite and simplify: ln(5𝑒 6 )
3) Expand the logarithm: log10
𝑦
2
4) Condense the logarithms: log 5 8 − log 5 𝑡