Learn on PengienVision, Algebra 2Chapter 6: Exponential and Logarithmic Functions

Lesson 5: Properties of Logarithms

In this Grade 11 enVision Algebra 2 lesson, students learn to apply the Product, Quotient, and Power Properties of Logarithms to expand and condense logarithmic expressions. Students also use the Change of Base Formula to evaluate logarithms in any base using a calculator. The lesson connects these properties to real-world applications such as calculating hydrogen ion concentration using the pH formula.

Section 1

Properties of logarithms

Property

If xx, yy, and b>0b > 0, and b≠1b \neq 1, then

  1. Product Rule: log⁡bxy=log⁡bx+log⁡by\log_b xy = \log_b x + \log_b y
  1. Quotient Rule: log⁡bxy=log⁡bx−log⁡by\log_b \frac{x}{y} = \log_b x - \log_b y

Section 2

Combining and expanding logarithms

Property

The properties of logarithms can be used to rewrite expressions. You can expand a single logarithm into a sum or difference of simpler logs, or combine multiple logs into a single, more complex logarithm.

Expand: log⁡b(xAyB)=Alog⁡bx+Blog⁡by\log_b(x^A y^B) = A \log_b x + B \log_b y

Combine: Alog⁡bx−Blog⁡by=log⁡b(xAyB)A \log_b x - B \log_b y = \log_b(\frac{x^A}{y^B})

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Properties of logarithms

Property

If xx, yy, and b>0b > 0, and b≠1b \neq 1, then

  1. Product Rule: log⁡bxy=log⁡bx+log⁡by\log_b xy = \log_b x + \log_b y
  1. Quotient Rule: log⁡bxy=log⁡bx−log⁡by\log_b \frac{x}{y} = \log_b x - \log_b y

Section 2

Combining and expanding logarithms

Property

The properties of logarithms can be used to rewrite expressions. You can expand a single logarithm into a sum or difference of simpler logs, or combine multiple logs into a single, more complex logarithm.

Expand: log⁡b(xAyB)=Alog⁡bx+Blog⁡by\log_b(x^A y^B) = A \log_b x + B \log_b y

Combine: Alog⁡bx−Blog⁡by=log⁡b(xAyB)A \log_b x - B \log_b y = \log_b(\frac{x^A}{y^B})