1. Logarithmic Properties
Exponential Function
y = bx
Logarithmic Function
x = by y = logbx
Exponential Form Logarithmic Form

2. Exponential Form Logarithmic Form
The logarithm is the exponent to which a base must be raised to give a power.
Exponential Form Logarithmic Form
53 = 125 
72 = 49 
5-2 = 1/25 
3-4 = 1/81 
9½ = 3 
36½ = 6 
be = P 

3. Exponential Form Logarithmic Form
The logarithm is the exponent to which a base must be raised to give a power.
Exponential Form Logarithmic Form
53 = 125 
72 = 49 
5-2 = 1/25 
3-4 = 1/81 
9½ = 3 
36½ = 6 
be = P 
3 = log5125
2 = log749
-2 = log5(1/25)
-4 = log3 (1/81)
½ = log93
½= log366
e = logbP

4. Exponential Form Logarithmic Form
The logarithm is the exponent to which a base must be raised to give a power.
Exponential Form Logarithmic Form
 4 = log381
 2 = log864
 -2 = log7(1/49)
 -4 = log2 (1/16)
 ¼ = log813
 1/5 = log322
 y = logax

5. Exponential Form Logarithmic Form
The logarithm is the exponent to which a base must be raised to give a power.
Exponential Form Logarithmic Form
 4 = log381
 2 = log864
 -2 = log7(1/49)
 -4 = log2 (1/16)
 ¼ = log813
 1/5 = log322
 y = logax
34 = 81
82 = 64
7-2 = 1/49
2-4 = 1/16
81¼ = 3
321/5 = 2
ay = x

6. Method #1
Method #2

7. Let 𝑦= 𝑙𝑜𝑔 2 32 𝑙𝑜𝑔 2 32= 𝑙𝑜𝑔 2 2 5 ∴ 𝑙𝑜𝑔 2 32=5 2 𝑦 =32 2 𝑦 = 2 5 𝑦=5
Method #1
Method #2
Write the Argument as a Power of the base of the log
Let 𝑦= 𝑙𝑜𝑔 2 32
𝑙𝑜𝑔 2 32= 𝑙𝑜𝑔
Change to Exponential Form
The Logarithm is the Exponent
∴ 𝑙𝑜𝑔 2 32=5
2 𝑦 =32
2 𝑦 = 2 5
Equate Exponents
𝑦=5
∴ 𝑙𝑜𝑔 2 32=5

8. Ex #2] Evaluate.
a) 𝑙𝑜𝑔
b) 𝑙𝑜𝑔 5 1
c) 𝑙𝑜𝑔

9. Ex #2] Evaluate. = 𝑙𝑜𝑔 5 5 0 = 𝑙𝑜𝑔 3 3 −4 = 𝑙𝑜𝑔 10 10 6 = 0 = - 4 = 6
Ex #2] Evaluate.
a) 𝑙𝑜𝑔
b) 𝑙𝑜𝑔 5 1
c) 𝑙𝑜𝑔
= 𝑙𝑜𝑔
= 𝑙𝑜𝑔 3 3 −4
= 𝑙𝑜𝑔
= 𝑙𝑜𝑔
= 0
= - 4
= 6

10. Method #1
Method #2

11. Let 𝑦= 𝑙𝑜𝑔 3 81 = 3 𝑙𝑜𝑔 3 3 4 3 𝑙𝑜𝑔 3 81 = 3 4 3 𝑦 =81 =81 3 𝑦 = 3 4
Method #1
Method #2
Let 𝑦= 𝑙𝑜𝑔 3 81
Write the argument of the Log as a Power
3 𝑙𝑜𝑔 3 81
= 3 𝑙𝑜𝑔
Change to Exponential Form
= 3 4
3 𝑦 =81
=81
3 𝑦 = 3 4
Equate Exponents
𝑦=4
∴ 3 𝑙𝑜𝑔 = 3 4
∴ 3 𝑙𝑜𝑔 =81

12. Ex#4] Evaluate.
a) 𝑙𝑜𝑔 4 64
b) 𝑙𝑜𝑔 7 125
c) 𝑙𝑜𝑔 4 16

13. Ex#4] Evaluate. a) 4 𝑙𝑜𝑔 4 64 b) 7 𝑙𝑜𝑔 7 125 c) 6 𝑙𝑜𝑔 4 16 =64 =125
Ex#4] Evaluate.
a) 𝑙𝑜𝑔 4 64
b) 𝑙𝑜𝑔 7 125
c) 𝑙𝑜𝑔 4 16
=64
=125
= 6 𝑙𝑜𝑔
= 6 2
=36

14. Your Turn… Read: “Example#2” on p.461-462
“In Summary” on p.465
Do: p #1cdf,2bde,3,4,5aef,6,9,10,19,20a
Corrections: #4d)approx 1.40