Algebra 2
16.1 Notes
16.1 (Day One) Properties of Logarithms
Date: ___________
Properties of Logarithms
Logarithmic expressions can be rewritten using one or more properties of logarithms.
Learning Target A: I can use the properties of logarithms.
Recall that a logarithm is the exponent to which a base must be raised in order to obtain a given number.
EXPONENTIAL EQUATION
LOGARITHMIC EQUATION
𝒃𝒙 = 𝒂
𝒍𝒐𝒈𝒃 𝒂 = 𝒙
𝒃 > 𝟎, 𝒃 ≠ 𝟏
A) Determine each of the following to identify the definition-based properties of logs:
If 𝒍𝒐𝒈𝒃 𝒃𝒎 = ___________,
It follows that 𝒍𝒐𝒈𝒃 𝒃𝟎 = ______, so 𝒍𝒐𝒈𝒃 𝟏 = _______.
Also, 𝒍𝒐𝒈𝒃 𝒃𝟏 = _____, so 𝒍𝒐𝒈𝒃 𝒃 = _________.
B) Just like we have exponent properties for powers of the same __________, we have log
properties for logs of the same base! Fill in the table to determine the properties of operations
with logs.
Product
Quotient
Power
Exponent Properties
Logarithm Properties
𝑎𝑚 ⋅ 𝑎𝑛 =
𝑙𝑜𝑔𝑏 𝑚𝑛 =
𝑎𝑚
𝑚
=
𝑙𝑜𝑔𝑏 𝑁 =
(𝑎𝑚 )𝑛 =
𝑙𝑜𝑔𝑏 𝑚𝑛 =
𝑎𝑛
Example 1. Expand each expression to be written in terms of log m and log n.
A) 𝑙𝑜𝑔𝑚2 𝑛5
B) 𝑙𝑜𝑔
3
√𝑚
𝑛4
1
Algebra 2
16.1 Notes
4
√𝑛
C) 𝑙𝑜𝑔𝑚4 𝑛2
D) 𝑙𝑜𝑔 𝑚2
Fill out the table with the properties of logarithms that you discovered on the first page.
Properties of Logarithms
For any positive numbers a, m, n, b, (𝒃 ≠ 𝟎), and c (𝒄 ≠ 𝟎), the following properties hold:
1. 𝑙𝑜𝑔𝑏 𝑏 𝑚 = 𝑚
Definition-Based Properties
2. 𝑙𝑜𝑔𝑏 1 = 0
3. 𝑙𝑜𝑔𝑏 𝑏 = 1
Product Property of Logarithms
𝑙𝑜𝑔𝑏 𝑚𝑛 =
Quotient Property of Logarithms
𝑙𝑜𝑔𝑏
Power Property of Logarithms
𝑙𝑜𝑔𝑏 𝑚𝑛 =
𝑚
=
𝑛
Example 2. Express each expression as a single logarithm. Evaluate without a calculator, if possible.
A) ln 18 − 2 ln 3 + ln 4
B) ln 25 + 4 ln 5 − ln 125
C) log 6 + log 11
D) 𝑙𝑜𝑔3 250 − 2𝑙𝑜𝑔3 10
1
E) 3 𝑙𝑜𝑔5 8 −
1
2
𝑙𝑜𝑔5 9
5
F) 2 𝑙𝑜𝑔7 16 +
2
3
𝑙𝑜𝑔7 125
2