Lesson 8-3 Logarithmic Functions as Inverses
Do now – Write in exponential form:
1) √3
3
4
2) √2
3) √21
3
4) √𝑎2
5) √𝑏
The inverse function of the exponential function with base b (i.e., 𝑓(𝑥) = 𝑏 𝑥 ) is called the logarithmic
function with base b.
Definition of the Logarithmic Function
For 𝑥 > 0 and 𝑏 > 0, 𝑏 ≠ 1,
𝑦 = 𝑙𝑜𝑔𝑏 𝑥 is equivalent to _____________
The function 𝑓(𝑥) = 𝑙𝑜𝑔𝑏 𝑥 is the __________________________________
Example 1 – Write each equation in its equivalent exponential form:
a) 2 = 𝑙𝑜𝑔5 𝑥
b) 𝑙𝑜𝑔𝑏 64 = 3
c) 𝑙𝑜𝑔3 7 = 𝑦
d) 3 = 𝑙𝑜𝑔7 𝑥
e) 2 = 𝑙𝑜𝑔𝑏 25
f) 𝑙𝑜𝑔4 26 = 𝑦
Example 2 – Write each equation in its equivalent logarithmic form:

a) 122 = 𝑥
b) 𝑏 3 = 8
d) 25 = 𝑥
e) 𝑏 3 = 27
c) 25 = 52
1 3
1
f) (2) = 8
To evaluate logarithms, you can write them in exponential form. A common logarithm
____________________________________________________________.
Example 3 – Evaluate:
1
5
a) 𝑙𝑜𝑔2 16
b) 𝑙𝑜𝑔7 49
c) 𝑙𝑜𝑔25 5
d) 𝑙𝑜𝑔2 √2
e) log 100
f) 𝑙𝑜𝑔5 125
1
g) 𝑙𝑜𝑔36 6
h) 𝑙𝑜𝑔3 √3
Graphing Logarithmic Functions
A logarithmic function is the inverse of an exponential function.
Example 4 – Graph each logarithmic function. Then find the domain and range.
a) 𝑦 = 𝑙𝑜𝑔2 𝑥
b) 𝑦 = 𝑙𝑜𝑔3 (𝑥 − 2) + 3
7
𝑐) 𝑦 = 𝑙𝑜𝑔4 (𝑥 + 3) − 1
Application Problems – Scientists use common logarithms to measure acidity, which increases as the
concentration of hydrogen ions in a substance increases. The pH of a substance equals – log[𝐻 + ], where
[𝐻 + ] is the concentration of hydrogen ions.
Example 5 – The pH of lemon juice is 2.3, while the pH of milk is 6.6. Find the concentration of hydrogen
ions in each substance. Which is more acidic?
Example 6 – Find the inverse of each function:
a) 𝑦 = 𝑙𝑜𝑔4 𝑥
b) 𝑦 = 𝑙𝑜𝑔2 2𝑥
c) 𝑦 = log(𝑥 + 1)
The domain of a Logarithmic function is _____________________________________
The Natural Base – 𝒆
An irrational number symbolized by the letter 𝒆. It is the base in many applied exponential functions. It is
1 𝑥
defined as (1 + 𝑥) as x approaches ∞. The function 𝑓(𝑥) = 𝑒 𝑥 is called the natural exponential function.
The Natural Logarithm – ln
Definition:
The function 𝑓(𝑥) = 𝑙𝑜𝑔𝑒 𝑥 is usually expressed as _______________
Like the domain of all logarithmic functions, the domain of the natural logarithm is ____________________
Example 7 – Find the domain of each function:
a) 𝑓(𝑥) = ln(3 − 𝑥)
b) 𝑔(𝑥) = ln(𝑥 − 3)2
c) ℎ(𝑥) = ln(𝑥 − 4)
d) 𝑔(𝑥) = log 𝑥 2
e) 𝑓(𝑥) = 𝑙𝑜𝑔3 (2𝑥 + 1)
f) 𝑦 = ln(6 − 4𝑥)
General Properties
1.
2.
3.
4.
𝑙𝑜𝑔𝑏 1 = 0
𝑙𝑜𝑔𝑏 𝑏 = 1
𝑙𝑜𝑔𝑏 𝑏 𝑥 = 𝑥
𝑏 𝑙𝑜𝑔𝑏𝑥 = 𝑥
Natural Logarithms
1. ln 1 = 0
2. ln 𝑒 = 1
3. ln 𝑒 𝑥 = 𝑥
4. 𝑒 𝑙𝑛𝑥 = 𝑥
Example 8 – Graph 𝑦 = 𝑒 𝑥 and 𝑦 = ln 𝑥 on the
same coordinate plane.
Homework:
Pages 442-443 #’s 6-25 and 53-61 (examples 1 – 3)
Pages 442-443 #’s 26, 27, 28, 35, 38, 40, 64, 68, and 69 (examples 4 – 6)