1. Logarithmic Functions
Today’s Objective:
I can write and evaluate logarithmic expressions.

2. 7 different ways
16,777,216= =
= 256 3
= 64 4
= 16 6
= 8 8
= 4 12
= 2 24
4 different ways
64= 64 1
= 8 2
= 4 3
= 2 6
5 different ways
4096=
= 64 2
= 16 3
= 8 4
= 4 6
= 2 12

3. Solving exponential equations using common bases
4 𝑥 = 1 16
4 𝑥 =32
5 𝑥 =125 2𝑥
2 = 2 5
Write each side with the same base.
4 𝑥 = 1 4
2𝑥=5 2
5 𝑥 = 5 3
𝑥= 5 2
4 𝑥 = 4 −2
Since bases are the same
exponents must be equal.
2 𝑥 =7
𝑥=−2
𝑥=3

4. Exponential & Logarithm Equations
Inverse
Exponential Equation
Logarithm Equation
𝑏 log 𝑏 𝑎 = 𝑎
Exponent/power
log 𝑏 𝑏 𝑎 = 𝑎
Base
Result
Common Log:
Logs with a base 10
Written log only – no base # needed
Calculator button: [LOG]
Read: log base b of a

5. 𝑎= 𝑏 𝑥 ↔ log 𝑏 𝑎 =𝑥 Write in logarithmic form
Write in exponential form
14= 5 𝑥
log =𝑥
log 14 5
log 3 8 =𝑥 𝑥
3 𝑥 =8 3
23= 𝑒 𝑥
log 𝑒 23 =𝑥
log =2𝑥
5 2𝑥 =34
6= 10 3𝑥+1
log 8 (6) +1=𝑥
log =3𝑥+1
log 8 6 =𝑥−1
log 6 =3𝑥+1
8 𝑥−1 =6

6. Solve each equation by using common base method
𝑎= 𝑏 𝑥 ↔ log 𝑏 𝑎 =𝑥
𝑥= log
log =2𝑥+3
log =𝑥−7
5 𝑥 =625
3 𝑥−7 =27
4 2𝑥+3 =64 3
3 𝑥−7 = 3
5 𝑥 = 5 4 3
4 2𝑥+3 =4
𝑥−7=3
2𝑥+3=3
𝑥=4
𝑥=0
𝑥=10