Properties of Logarithmic Functions
Learning Objective(s)
· Express the logarithm of a product as a sum of logarithms.
· Express the logarithm of a quotient as a difference.
· Express the logarithm of a power as a product.
· Simplify logarithmic expressions.
Introduction
Throughout your study of algebra, you have come across many properties—such as the commutative, associative, and distributive properties. These properties help you take a complicated expression or equation and simplify it.
The same is true with logarithms. There are a number of properties that will help you simplify complex logarithmic expressions. Since logarithms are so closely related to exponential expressions, it is not surprising that the properties of logarithms are very similar to the properties of exponents. As a quick refresher, here are the exponent properties.
Properties of Exponents
Product of powers: Quotient of powers: Power of a power:

One important but basic property of logarithms is log_{b }b^{x} = x. This makes sense when you convert the statement to the equivalent exponential equation. The result? b^{x} = b^{x}.
Let’s find the value of y in. Remember , so means and y must be 2, which means . You will get the same answer that equals 2 by using the property that log_{b }b^{x} = x.
Remember that the properties of exponents and logarithms are very similar. With exponents, to multiply two numbers with the same base, you add the exponents. With logarithms, the logarithm of a product is the sum of the logarithms.
Logarithm of a Product
The logarithm of a product is the sum of the logarithms: log_{b }(MN) = log_{b }M + log_{b }N

Let’s try the following example.
Example  
Problem  Use the product property to rewrite .  

 Use the product property to write as a sum. 

 Simplify each addend, if possible. In this case, you can simplify both addends. Rewrite log_{2 }4 as log_{2 }2^{2}and log_{2 }8 as log_{2 }2^{3}, then use the property log_{b }b^{x} = x. Or, rewrite log_{2 }4 = y as 2^{y }= 4 to find y = 2, and log_{2 }8 = y as 2^{y }= 8 to find y = 3. Use whatever method makes sense to you. 
Answer 


Another way to simplify would be to multiply 4 and 8 as a first step.
You get the same answer as in the example!
Notice the similarity to the exponent property: b^{m}b^{n} = b^{m}^{ + n}, while log_{b }(MN) = log_{b }M + log_{b }N. In both cases, a product becomes a sum.
Example  
Problem  Use the product property to rewrite log_{3 }(9x).  
 log_{3 }(9x) = log_{3 }9 + log_{3 }x  Use the product property to write as a sum. 
 log_{3 }9 + log_{3 }x = log_{3 }3^{2} + log_{3 }x = 2 + log_{3 }x  Simplify each addend, if possible. In this case, you can simplify log_{3 }9 but not log_{3 }x. Rewrite log_{3 }9 as log_{3 }3^{2}, then use the property log_{b }b^{x} = x. Or, simplify log_{3 }9 by converting log_{3 }9 = y to 3^{y} = 9 and finding that y = 2.
Use whatever method makes sense to you. 
Answer  log_{3}(9x) = 2 + log_{3 }x 

If the product has many factors, you just add the individual logarithms:
log_{b }(ABCD) = log_{b }A + log_{b }B + log_{b }C + log_{b }D.
Rewrite log_{2 }8a, then simplify.
A) 3 log_{2 }a B) log_{2 }3a C) log_{2 }(3 + a) D) 3 + log_{2 }a

You can use the similarity between the properties of exponents and logarithms to find the property for the logarithm of a quotient. With exponents, to multiply two numbers with the same base, you add the exponents. To divide two numbers with the same base, you subtract the exponents. What do you think the property for the logarithm of a quotient will look like?
As you may have suspected, the logarithm of a quotient is the difference of the logarithms.
Logarithm of a Quotient

With both properties: and, a quotient becomes a difference.
Example 
 
Problem  Use the quotient property to rewrite . 
 
 log_{2 } = log_{2 }x – log_{2 }2  Use the quotient property to rewrite as a difference.  
Answer  = log_{2 }x – 1  The first expression can’t be simplified further. However, the second expression can be simplified. What exponent on the base (2) gives a result of 2? Since 2^{1} = 2, you know log_{2 }2 = 1.  
Which of these is equivalent to: .
A) 4 – log_{3 }a B) C) log_{3 }(4 – a) D)

The remaining exponent property was power of a power: . The similarity with the logarithm of a power is a little harder to see.
Logarithm of a Power

With both properties, and, the power “n” becomes a factor.
Example  
Problem  Use the power property to simplify log_{3 }9^{4}.  
 log_{3 }9^{4} = 4 log_{3 }9
 You could find 9^{4}, but that wouldn’t make it easier to simplify the logarithm. Use the power property to rewrite log_{3 }9^{4} as 4log_{3 }9. 
 4 log_{3 }9 = 4•2  You may be able to recognize by now that since 3^{2} = 9, log_{3 }9 = 2. 
Answer  log_{3 }9^{4 = }8  Multiply the factors. 
Notice in this case that you also could have simplified it by rewriting it as 3 to a power: log_{3 }9^{4} = log_{3 }(3^{2})^{4}. Using exponent properties, this is log_{3 }3^{8} and by the property log_{b }b^{x} = x, this must be 8!
Example  
Problem  Use the properties of logarithms to rewrite log_{4 }64^{x}.  

 Use the power property to rewrite as .
Rewrite as, then use the property to simplify . Or, you may be able to recognize by now that since 4^{3} = 64, log_{4}64 = 3.  


 
Answer  log_{4 }64^{x} = 3x  Multiply the factors.  
Which of these is equivalent to: log_{2 }x^{8}.
A) log_{2 }3x B) 8 log_{2 }x C) log_{2 }8x D) 3 log_{2 }x

The properties can be combined to simplify more complicated expressions involving logarithms.
Example  
Problem  Use the properties of logarithms to expand into four simpler terms.  




 Use the quotient property to rewrite as a difference of logarithms. 

 Now you have two logarithms, each with a product. Apply the product rule to each.
Be careful with the subtraction! Since all of log_{10 }cd is subtracted, you have to subtract both parts of the term, (log_{10 }c + log_{10 }d). 
Answer  = log_{10 }a + log_{10 }b – log_{10 }c – log_{10 }d 

Example  
Problem  Simplify log_{6 }(ab)^{4}, writing it as two separate terms.  




 Use the power property to rewrite log_{6 }(ab)^{4} as 4 log_{6 }(ab).
You are taking the log of a product, so apply the product property.
Be careful: the value 4 is multiplied by the whole logarithm, so use parentheses when you rewrite log_{6 }(ab) as (log_{6 }a + log_{6 }b) 
Answer  log_{6 }(ab)^{4 } = 4 log_{6 }a + 4 log_{6 }b
 Use the distributive property. 
Simplify log_{3 }x^{2}y.
A) 2(log_{3 }x + log_{3 }y) B) log_{3 }x^{2} + log_{3 }y C) 2 log_{3 }xy D) 2 log_{3 }x + log_{3 }y

Summary
Like exponents, logarithms have properties that allow you to simplify logarithms when their inputs are a product, a quotient, or a value taken to a power. The properties of exponents and the properties of logarithms have similar forms.
 Exponents  Logarithms 
Product Property 


Quotient Property 


Power Property 


Notice how the product property leads to addition, the quotient property leads to subtraction, and the power property leads to multiplication for both exponents and logarithms.