Adding & Subtracting Logarithms

Adding & Subtracting Logarithms
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  • 0:01 Logarithms
  • 2:02 Adding Logarithms
  • 3:11 Subtracting Logarithms
  • 4:41 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Logarithms are a fascinating subject in mathematics. They take on many beautiful patterns; two of these happen when we are adding and subtracting logarithms. Let's explore these patterns and deepen our understanding of logarithms in general.

Logarithms

Indications of logarithms go back as far as 8th century India; however, their discovery and use in mathematics is credited to John Napier, a 17th century Scotsman.

Exponents and logarithms go hand in hand. In fact, logarithms are exponents.

A Logarithm
logarithm2

When we read this expression, we say log base b of x equals y. We call b the base of the logarithm, x the power of the logarithm, and y the exponent of the logarithm. Log base b of x equals y represents the number, or exponent 'y', that we have to raise b to in order to get x. A good way to remember this is to use a process called 'Rock and Roll' that makes it easy to go from logarithmic form to exponential form. Let's see how this works:

Logb^x = y is equivalent to b^y = x

We can change logarithmic form to exponential form by starting with the base, 'b.' To see what we raise 'b' to, we rock all the way forward to 'y.' Now we have 'b' to the 'y' power. To find out what this equals, you roll back to 'x.' Now we have 'b' to the 'y' power equals 'x.'


Rock and Roll from logarithmic form to exponential form
logarithm3


For example, suppose we are asked to find log 3 9. We set log 3 9 equal to a variable, say y, and then we use rock and roll to see that y represents the number we raise 3 to in order to get 9.


Rock and Roll log 3 9
logarithm4


We know that we raise 3 to the power of 2 to get 9, so log 3 9 = 2.

Working with logarithms is actually quite entertaining because they take on many fun patterns, like the 'Rock and Roll' pattern that we just saw. Two very important patterns logarithms follow happen when we add or subtract logarithms with common bases. These are called the multiplication rule of logarithms and the division rule of logarithms.

Adding Logarithms

The multiplication rule of logarithms applies when we are adding two logarithms together that have the same base.


The Multiplication Rule of Logarithms
logarithm5


In words, when we add log base b of M to log base b of N, it is just the same as taking log base b of M times N. This rule comes in very handy when we want to add two logarithms together that have the same base, but are not easy to calculate individually.

For example, suppose we want to find log 4 2 + log 4 32. Both of these logarithms are not easy to calculate individually. You may not know what power you would need to raise 4 to in order to get 2 or to get 32 off the top of your head. However, we notice that both logarithms have base 4, and we are adding them together, so we can apply the multiplication rule for logarithms! Let's see what happens when we do!

log 4 2 + log 4 32 Apply the multiplication rule
log 4 (2*32) Simplify
log 4 64 The number we raise 4 to in order to get 64
log 4 64 = 3 Thus, log 4 2 + log 4 32 = 3

We see that log 4 2 + log 4 32 = 3. This rule really comes in handy when calculating logarithms and simplifying addition of logarithms with the same base.

Subtracting Logarithms

We also have a rule that helps when we are subtracting logarithms with the same base. This rule is the division rule of logarithms, which is as follows.


The Division Rule of Logarithms
logarithm6


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