Lesson 5

Lesson 5: Laws of Logarithms

 

Focus

 

Most people can hear a wide range of sound levels. As the intensity of a sound increases, you perceive the sound as getting louder. It is possible to determine “how many times as loud” one sound is compared to another using the equation S = 10 log I1 − 10 log I0, where I1 and I0 are the two sound intensities. In this lesson you will learn to simplify logarithmic expressions such as this one.

 

Lesson Outcome

 

At the end of this lesson you will be able to use the laws of logarithms to manipulate expressions.

 

Lesson Question

 

You will investigate the following question: How can the laws of logarithms be used to manipulate expressions?

 

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 5 Assignment (Download the Lesson 5 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Formula Sheet
• work under Project Connection

Discover

 

So far, you have seen that logarithms and exponents are closely related.

 

 

 

 

There are rules that can be used to combine powers with the same base. The next activity explores a related set of rules for logarithms.

 

Try This 1

1. Complete a table like the following:
   
   

   log 100 + log 1000	= 2 + 3	= 5	= log ____
   log 10 + log 100	=	=	= log ____
   log 10 000 + log 0.01	=	=	= log ____

2.  
   1. Explain any patterns you see in the table.
   2. Use your pattern to predict a rule for simplifying log M + log N.
3. Check that your pattern works with numbers that are not powers of 10 by completing a table like the following using a calculator:
   
   

   log 12 + log 3 =	log 36 =
   log 5 + log 7 =	log ____ =
   log 4 + log 9 =	log ____ =

4. Predict a rule for simplifying log M − log N.

5. Complete a table like the following using a calculator to test your rule:
   
   

   log 18 − log 6 =	log ____ =
   log 8 − log 2 =	log ____ =
   log 10 − log 5 =	log ____ =

Save your responses in your course folder.

 

Share 1

 

With a partner or group, describe any patterns you see between the addition and subtraction rules for logarithms you developed in Try This 1 and the following exponent laws:

 

(bm)(bn) = bm+n

 

If required, save a summary of your discussion in your course folder.

Explore

 

In Discover, you may have found that logarithms can be added or subtracted using laws similar to

• log M + log N = log(M × N)
• 

These rules can be extended to any base to give

• logb M + logb N = logb(M × N)
• 

Notice that the bases must be the same to use these laws. The same restrictions apply to these rules as to other logarithms: b, M, and N are real numbers greater than 0, and b ≠ 1.

 

The following table provides an explanation of how the logarithmic laws are related to exponential laws. Remembering that a logarithm is an exponent may help you interpret the table.

 

Law of Logarithms	Law Expressed Mathematically	Pattern Explained by the Law of Powers
product law of logarithms	logb (M × N) = logb M + logb N	In the product law of powers, (bm)(bn) = bm + n, the exponents are added. In the product law of logarithms, logb (M × N) = logb M + logb N, when you are multiplying two terms in a single logarithm, it is the same as adding the logarithms of each term.
quotient law of logarithms		In the quotient law of powers, , the exponents are subtracted. In the quotient law of logarithms, , when you are dividing two terms in a single logarithm, it is the same as subtracting the logarithms of each term.

So far, you have seen the product and quotient laws of logarithms. There is a third law that is commonly used with logarithms. The next activity explores this rule.

 

Try This 2

1.  Complete a table like the following:
   
   

   log 2 = log 21 =	1 × log 2 =
   log 4 = log 22 =	2 × log 2 =
   log 8 = log 23 =	3 × log 2 =
   log 16 = log 24 =	4 × log 2 =

2. 1. What pattern do you notice from the table?
   2. Use your rule to predict an alternate way of writing log 54.
   3. Use your rule to predict an alternate way of writing 6 log 2.
3. Does your rule work for other bases, such as log4? Explain.

Save your responses in your course folder.

 

Share 2

 

With a partner or group, describe any patterns you see between your rule from Try This 2 and the exponent law (bm)n = bm × n.

 

If required, save a summary of your discussion in your course folder.

In Try This 2, you may have found that the logarithm of a power can be rewritten as the product of a number and a logarithm using the law logb (Mn)= n logb M. The following explains the relationship between this law and the power law of powers:

 

power law of logarithms

logb (Mn)= n logb M

In the product law of powers, (bm)n = bm × n, the exponents are multiplied. In the power law of logarithms, logb (Mn)= n logb M, when you have a power in a single logarithm, it is the same as multiplying the exponent to the logarithm of the base of the power.

Here are some numeric examples of the power law of logarithms:

 

The power law of logarithms is useful when trying to solve exponential equations. Lesson 6 will explore this idea.

 

Self-Check 2

Connect

 

Lesson 5 Assignment

Lesson 5 Summary

 

In this lesson you learned the following three laws of logarithms:

• logb M + logb N = logb(M × N)
• 
• logb (MN) = n logb M

These laws can be used to manipulate expressions to a more desirable form. In the next lesson you will see how these can be used to help solve exponential equations.