Expanding logarithms

The examples below will teach you about expanding logarithms using the properties of logarithms.

Expanding logarithms is the inverse of simplifying logarithms. You are trying to express a single logarithm into many individual logarithms.

Example #1

Expand log5 (3xy)

Using the product property, log5 (3xy) = log5 3 + log5 x + log5 y

Example #2

Expand log2 (5x / 10y)

Using a combination of the product property and the quotient property,

log2 (5x / 10y)  = log2 (5x) -  log2 (10y)

                        = log2 (5) +  log2 (x)  -  [ log2 (10) +  log2 (y) ]

                        = log2 (5) +  log2 (x)  -   log2 (10) -  log2 (y) 

Example #3

Expand log (a2b3)

Using a combination of the product property and the power property,

log10 (a2b3)    = log10 a2 + log10 b3

                       = 2log10 a + 2log10


Example #4

Expand log3 [ √ (36x3y) ]

Using a combination of the product property and the power property,

log3 [ √ (36x3y) ]   =  log3 [  (36x3y)1/2 ]

                              = 1/2 log3 (36x3y)

                              = 1/2 [ log3 (36) +  log3 (x3) + log3 (y) ]

                              = 1/2 [ log3 (36) +  3log3 (x) + log3 (y) ]

                              = 1/2 log3 (36) +  3/2 log3 (x) + 1/2 log3 (y)




Recent Articles

  1. Why Multiply before Adding when doing PEMDAS

    Oct 02, 19 04:34 PM

    Why multiply before adding? The common sense behind doing multiplication before addition

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

            Follow me on Pinterest

Math quizzes

 Recommended

Scientific Notation Quiz

Graphing Slope Quiz

Adding and Subtracting Matrices Quiz  

Factoring Trinomials Quiz 

Solving Absolute Value Equations Quiz  

Order of Operations Quiz

Types of angles quiz

Tough algebra word problems

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Recent Articles

  1. Why Multiply before Adding when doing PEMDAS

    Oct 02, 19 04:34 PM

    Why multiply before adding? The common sense behind doing multiplication before addition

    Read More

K-12 math tests


Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. 

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.