1. Property #1 – Product Property log a (cd) =
Algebra 2/Trigonometry Name: __________________________
Section 8.5 Fill In Notes Date: ___________________________
8.5 Properties of Logarithms
Most calculators only have two types of log keys:
1.) Common Logarithms (Base ____)
2.) Natural Logarithms (Base ____)
To evaluate logarithms to other bases we need to use the CHANGE OF BASE FORMULA!
log a x can be converted to a different base as follows:
BASE b
BASE 10
BASE e
log a x =
Changing Bases using Common & Natural Logarithms
Common Log
Natural Log
log =
log =
log =
log =
Property #1 – Product Property
log a (cd) =
Log Properties:
How well do you actually know them?
log =___
Examples:
log = ln (xyz) =

2. Property #2 – Quotient Property log a c d =
Log Properties:
How well do you actually know them?
log =___
Examples:
log = log xy zw =
Property #3 – Power Property
log a c w =
Log Properties:
How well do you actually know them?
ln e =___
Examples:
log = log =
log = ln56=
Simplify each expression.
log − log 5 3
ln 𝑒 6 𝑒 2
log log 4 32

3. Condense each expression into one logarithm.
Log Properties:
How well do you actually know them?
log 104=___
Condense each expression into one logarithm.
1.) log 𝑥+3log⁡(𝑥+1) 2.) 2 ln(x+2) – lnx
3.) log 3 (𝑥+2) − log 3 𝑥 −2 log 3 𝑥
Log Properties:
How well do you actually know them?
ln e=___
Expand each expression.
1.) log 4 5 𝑥 3 𝑦
2.) ln 3𝑥−5 7
3.) log(4x3y5z2)

4. Algebra 2/Trigonometry Name: __________________________
Section 8.6 Fill In Notes Date: ___________________________
All of these functions are one-to-one. (They pass the ________________________).
This means that every x value has _______________________________.
Solving Exponential Equations Using the One-to-One Property:
Example 1: Example 2:
Example 3: Example 4:
Solving Exponential Equations w/o One-to-One:
5. 3x = 30
(¼)x = 60
(2x) = 42
8. ex = 7 9.
5 - 3ex = 2
10.
2(32t-5) - 4 = 11
Example 11
e2x - 7ex = -12
Example 12
e2x - 4ex – 5 = 0

5. Solving Logarithmic Equations:
Recall: If _____________________________________________ .
Solving Logarithmic Equations Using the One-to-One Property:
Example 1: Example 2: Example 3:
Solving Logarithmic Equations w/o One-to-One: 4)
lnx = -3 5)
logx = -1
6) 2log5(3x) = 4
7) lnx – ln3 = 0 8)
log3(2x + 1) + log3(2) = log3(5x) 9)
log6(3x + 14) – log6(5) = log6(2x)
10)
log(5x) + log(x – 1) = 2
11)
log4x – log4(x – 1) = ½

6. Algebra 2/Trigonometry Name: __________________________
Section 8.6 Fill In Notes Date: ___________________________
Solving Logarithmic Equations, continued:
logx + log(x + 4) = 1
Application Example 1: If you had $2000 to invest in an account with an APR of 6% (compounded monthly), how many years would it take for the account to be worth $5000? Round to the nearest hundredths of a year.
Application Example 2: You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? Round to the nearest hundredths of a year.