1. Unit 8 [7-3 in text] Logarithmic Functions
Today’s Objective:
I can write and evaluate logarithmic expressions.

2. 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

3. 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

4. Solving using logs – any base

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

7. Find the inverse of each function
𝑎= 𝑏 𝑥 ↔ log 𝑏 𝑎 =𝑥
𝑦= 5log 7 (𝑥+4)
𝑦= 3 𝑥 4
𝑦= 3 4𝑥 +2
𝑥= 5log 7 (𝑦+4)
𝑥= 3 4𝑦 +2
𝑥 5 = log 7 (𝑦+4)
𝑥= 3 𝑦 4
𝑥−2= 3 4𝑦
log 3 (𝑥−2) =4𝑦
7 𝑥 5 =𝑦+4
4𝑥= 3 𝑦
log 3 (𝑥−2) 4 =𝑦
7 𝑥 5 −4=𝑦
log 3 4𝑥 =𝑦