1. Lesson 9 – 3 Logarithmic Functions
Pre-calculus

2. Learning Objective To solve log equations Graph log functions
To evaluate log expressions

3. Logarithmic Function 𝑓 𝑥 = 𝑏 𝑥 is one–to–one so it has an inverse
Logarithmic Function – The inverse of an exponential function
For positive real numbers 𝑥 and 𝑏,
𝑏>0 and 𝑏≠1,
𝑦= 𝑙𝑜𝑔 𝑏 𝑥 iff 𝑥= 𝑏 𝑦
D = 𝑥>0 D =ℛ
inverses
R = ℛ R =𝑦>0
Do “around the world”
start with base
𝑙𝑜𝑔 𝑏 𝑥=y
⇒ 𝑏 𝑦 =𝑥

4. Logarithmic Equation Solve for x 1. 𝑙𝑜𝑔 7 𝑥=2 7 2 =𝑥 49=𝑥
1. 𝑙𝑜𝑔 7 𝑥=2
7 2 =𝑥
49=𝑥
2. 𝑙𝑜𝑔 =𝑥
2 𝑥 = 1 16
2 𝑥 = 2 −4
𝑥=−4
3. 𝑙𝑜𝑔 𝑥 81=4
( ) ( ) 1 4
𝑥 4 =81
𝑥=3

5. Solve for x
Check – up
1. 𝑙𝑜𝑔 𝑥 125=−3
𝑥 −3 =125
𝑥= 1 5

6. Graph. Find Domain, Range, x– & y–int, asymptote, increasing, or decreasing.
Logarithmic Function
4. 𝑓 𝑥 = 𝑙𝑜𝑔 2 𝑥
 𝑦= 𝑙𝑜𝑔 2 𝑥
𝑥 𝑦
2 𝑦 =𝑥
2 −2 = 1 4 −2 −1 1 2
2 −1 = 1 2
Plug values into y
2 0 =1
2 1 =2
2 2 =4
D: 𝑥>0
R: ℛ
x–int: (1, 0)
y–int: none
Asym: x = 0
increasing

7. Graph. Find Domain, Range, x– & y–int, asymptote, increasing, or decreasing.
Logarithmic Function
5. 𝑓 𝑥 = 𝑙𝑜𝑔 𝑥
 𝑦= 𝑙𝑜𝑔 𝑥
𝑥 𝑦
1 2 𝑦 =𝑥
2 2 =4 −2 −1 1 2
Plug values into y
2 1 =2
2 0 =1
2 −1 = 1 2
2 −2 = 1 4
D: 𝑥>0
R: ℛ
x–int: (1, 0)
y–int: none
Asym: x = 0
decreasing

8. Logarithmic Function Summary: In the function 𝑓 𝑥 = 𝑏 𝑥
if 𝑏>1, 𝑓(𝑥) increases
if 0<𝑏<1, 𝑓(𝑥) decreases

9. Graph. Find Domain, Range, x– & y–int, asymptote, increasing, or decreasing.
Check – up
2. 𝑦= 𝑙𝑜𝑔 4 𝑥
D: 𝑥>0
R: ℛ
x–int: (1, 0)
y–int: none
Asym: x = 0
decreasing

10. The base of a log function can be any positive number except 1.
Basic Log Facts
But, there are two popular & powerful common bases.
Common Log: 𝒍𝒐𝒈 𝟏𝟎 𝒙  𝐥𝐨𝐠 𝒙
Natural Log: 𝒍𝒐𝒈 𝒆 𝒙  𝐥𝐧 𝒙
These have MANY applications to science & engineering (we’ll see tomorrow)
Basic Log Facts:
𝑙𝑜𝑔 𝑏 𝑏 𝑥 =𝑥
𝑙𝑜𝑔 𝑏 𝑏=1
𝑏 𝑙𝑜𝑔 𝑏 𝑥 =𝑥
𝑙𝑜𝑔 𝑏 1=0

11. Logarithmic Equation Simplify Each Expression 6. 𝑙𝑜𝑔 6 1 9. log 1 100
6. 𝑙𝑜𝑔 6 1
9. log =0
(log fact!)
= 𝑙𝑜𝑔 −2
=−2
7. 𝑙𝑜𝑔 3 81
= 𝑙𝑜𝑔
10. 𝑙𝑜𝑔 2 (−8) =4
Undefined
Why???
8. ln 𝑒 3
2 𝑥 =−8
not possible!!
= 𝑙𝑜𝑔 𝑒 𝑒 3
11. 𝑒 ln 6 =3
=𝑒 𝑙𝑜𝑔 𝑒 6 =6

12. Logarithmic Equation Simplify or Solve Each Expression 12. ln 1 𝑒 7
3𝑥+15=12
3𝑥= -3 −7
𝑥=−1
14. 𝑙𝑜𝑔 𝑥−2 =19
3𝑥−2=19
3𝑥=21
𝑥=7

13. Solve
Check – up
3. ln 𝑒 7𝑥−3 =25
𝑥=4

14. Lesson 9–3 Logarithmic Functions WS
Assignment
Lesson 9–3 Logarithmic Functions WS