1. 16.1 – Properties of Logarithms

2. where 𝑚 and 𝑛 are positive numbers and 𝑏≠1
Properties
Definition-based Properties:
𝒍𝒐𝒈 𝒃 𝒃 𝒎 =𝒎
𝒍𝒐𝒈 𝒃 𝟏 =𝟎
𝒍𝒐𝒈 𝒃 𝒃 =𝟏
Product Property: log 𝑏 𝒎𝒏 = log 𝑏 𝒎 + log 𝑏 𝒏
Quotient Property: log 𝑏 𝒎 𝒏 = log 𝑏 𝒎 − log 𝑏 𝒏
Power Property: log 𝑏 𝒎 𝒏 =𝒏 log 𝑏 𝒎
where 𝑚 and 𝑛 are positive numbers and 𝑏≠1

3. Proof of Product Property:

4. Ex:
What is each expression written as a single logarithm?
log − log b) log log
c) log

5. Ex: What is each expression written as a single logarithm?
5 log 3 𝑥 log 3 𝑦 b) log 𝑥 −7 log 𝑦
log 3 𝑥 log 3 𝑦 by Power Property log 𝑥 − log 𝑦 7 by Power Property
= log 3 𝑥 5 𝑦 by Product Property = log 𝑥 𝑦 by Quotient Property

6. Ex: What is each logarithm expanded? log 3 ( 𝑥 2 𝑦 7 ) log 8 𝑥 𝑦 3
a) log 3 𝑥 log 3 𝑦 7 by Product Property
=2 log 3 𝑥 +7 log 3 𝑦 by Power Property
b) log 8 𝑥 − log 8 𝑦 3 by Quotient Property
= log 8 𝑥 − log 8 𝑦 3 by inside/outside
= log 8 𝑥 −3 log 8 𝑦 by Power Property
c) log 4 4𝑥 − log 4 𝑦 9 by Quotient Property
= log log 4 𝑥 − log 4 𝑦 9 by Product Property
=1+ log 4 𝑥 −9 log 4 𝑦 by Power Property and since 4 1 =4

7. Evaluating Logarithmic Functions Using a Scientific Calculator
You can use a scientific calculator to find the logarithm of any positive number x when the logarithm’s base is either 10 or e.
When the base is 10, you are finding what is called the common logarithm of x, and you use the calculator’s key because log 10 𝑥 is also written as log 𝑥 .
When the base is e, you are finding what is called the natural logarithm of x, and you use the calculator’s key because log 𝑒 𝑥 is also written as ln 𝑥 .

8. Note: The “log” function on your calculators finds log 10 of a number.
To evaluate a logarithm with ANY base, use the Change of Base Formula:
log 𝒃 𝒎 = log 𝑐 𝒎 log 𝑐 𝒃

9. Ex: Use the Change of Base Formula to evaluate each expression.
log b) log
log 3 6 = log log =1.63
Method 1: log = log log = . 6
Method 2: log = log log = 2 3 because 4 2 =16 and 4 3 =64

10. On Your Own:
Use the Change of Base Formula to evaluate each expression.
log b) log 3 9
log 4 5 = log log =1.16
Method 1: log 3 9 = log log =2
Method 2: log 3 9 = log log = 2 1 =2 because 3 2 =9 and 3 1 =3

11. Ex:
A certain radioactive material decays according to the law 𝐴= 𝐴 0 𝑒 −0.021𝑡 , where 𝐴 0 is the initial amount present and 𝐴 is the amount present in 𝑡 years. What is the halflife of this material? Round the answer to two decimal places.
66.01
95.24
33.01
impossible to determine without knowing 𝐴 0
Answer: (D)

12. Ex:
The population of the United States in 2012 was million. If the population increases exponentially at an average rate of 1% each year, how long will it take for the population to double?
𝑦=𝑎 (1+𝑟) 𝑡 by exponential growth formula
627.8=313.9 (1+0.01) 𝑡 plugging in the values and doubling 313.9
2= (1.01) 𝑡 dividing by on both sides
log =𝑡 rewriting it as a log
𝑡= log log Change of Base
𝑡=69.66≈70 years calculator
Year = years

13. Ex:
A bank account earns 6% annual interest compounded annually. The balance B of the account after t years is given by the equation 𝐵= 𝐵 0 (1.06) 𝑡 , where 𝐵 0 is the starting balance. If the account starts with a balance of $250, how long will it take to triple the balance of the account?
𝐵= 𝐵 0 (1.06) 𝑡
750=250 (1.06) 𝑡 plugging in 250 and tripling 250
3= (1.06) 𝑡 dividing by 250 on both sides
log =𝑡 rewriting it as a log
𝑡= log log Change of Base
𝑡=18.85 years calculator

14. Reflect:
State each of the Product, Quotient, and Power Properties of Logarithms in a simple sentence. ____________________________________________________________________________________________________________________________________________________________________________________