18.2.2: Properties of Logarithmic Functions
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- 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.
Product of powers: bmbn=bm+n
Quotient of powers: bmbn=bm−n
Power of a power: (bm)n=bmn
One important but basic property of logarithms is logbbx=x. This makes sense when you convert the statement to the equivalent exponential equation. The result? bx=bx.
Let’s find the value of y in log332=y. Remember logbx=y⇔by=x, so log332=y means 3y=32 and y must be 2, which means log332=2. You will get the same answer that log332 equals 2 by using the property that logbbx=x.
Logarithm of a Product
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.
The logarithm of a product is the sum of the logarithms: logb(MN)=logbM+logbN
Let’s try the following example.
Problem: Use the product property to rewrite log2(4⋅8).
Answer
Use the product property to write as a sum.
log2(4⋅8)=log24+log28
Simplify each addend, if possible. In this case, you can simplify both addends. First rewrite log24 as log222 and log28 as log223, and then use the property logbbx=x.
Or, rewrite log24=y as 2y=4 to find y=2, and log28=y as 2y=8 to find y=3.
Use whatever method makes sense to you.
log24+log28=log222+log223=2+3=5
log2(4⋅8)=5
Another way to simplify log2(4⋅8) would be to multiply 4 and 8 as a first step.
log2(4⋅8)=log232=5 because 25=32
You get the same answerlog2(4⋅8)=5 as in the example!
Notice the similarity to the exponent property: bmbn=bm+n, while logb(MN)=logbM+logbN. In both cases, a product becomes a sum.
Problem: Use the product property to rewrite log3(9x).
Answer
Use the product property to write as a sum.
log3(9x)=log39+log3x
Simplify each addend, if possible. In this case, you can simplify log39 but not log3x. Rewrite \(\log _{3} 9\ as log332 and then use the property logbbx=x.
Or, simplify log39 by converting log39=y to 3y=9 and finding that y=2. Use whatever method makes sense to you.
log39+log3x=log332+log3x=2+log3x
log3(9x)=2+log3x
If the product has many factors, you just add the individual logarithms:
logb(ABCD)=logbA+logbB+logbC+logbD
Rewrite log28a, then simplify.
- 3log2a
- log23a
- log2(3+a)
- 3+log2a
- Answer
-
- Incorrect. The individual logarithms must be added, not multiplied. The correct answer is 3+log2a.
- Incorrect. You found that log28=3, but you must first apply the logarithm of a product property. The correct answer is 3+log2a.
- Incorrect. The logarithm of a product property says you separate the 8 and a into separate logarithms. The correct answer is 3+log2a.
- Correct. The logarithm of a product property says log28a=log28+log2a, and log28=3.
Logarithm of a Quotient
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.
logb(MN)=logbM−logbN
With both properties: bmbn=bm−n and logb(MN)=logbM−logbN, a quotient becomes a difference.
Problem: Use the quotient property to rewrite logx(x2).
Answer
Use the quotient property to rewrite as a difference.
log2(x2)=log2x−log22
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 21=2, you know log22=1.
log2(x2)=log2x−1
Which of these is equivalent to log3(81a)?
- 4−log3a
- 4log3a
- log3(4−a)
- log3(4a)
- Answer
-
- Correct. The logarithm of a quotient property states log3(81a)=log381−log3a, and log381=4.
- Incorrect. The individual logarithms must be subtracted, not divided. The correct answer is 4−log3a.
- Incorrect. The logarithm of a quotient property says you separate the 81 and a into separate logarithms. The correct answer is 4−log3a.
- Incorrect. You found that log381=4, but you must first apply the logarithm of the quotient property. The correct answer is 4−log3a.
Logarithm of a Power
The remaining exponent property was power of a power: (bm)n=bmn. The similarity with the logarithm of a power is a little harder to see.
logbMn=nlogbM
With both properties, (bm)n=bmn and logbMn=nlogbM, the power “n” becomes a factor.
Problem: Use the power property to simplify log394.
Answer
You could find 94 but that wouldn’t make it easier to simplify the logarithm. Use the power property to rewrite log394 as 4log39.
log394=4log39
You may be able to recognize by now that since 32=9, log39=2.
4log39=4⋅2
Multiply the factors.
log394=8
Notice in this case that you also could have simplified it by rewriting it as 3 to a power: log394=log3(32)4. Using exponent properties, this is log338 and by the property logbbx=x, this must be 8!
Problem: Use the properties of logarithms to rewrite log464x.
Answer
Use the power property to rewrite log464x as xlog464.
64=4⋅4⋅4=43
Rewrite log464 as log443, then use the property logbbx=x to simplify log443.
Or, you may be able to recognize by now that since 43=64, log464=3.
log464x=xlog464=xlog443=x⋅3
Multiply the factors.
log464x=3x
Which of these is equivalent to log2x8?
- log23x
- 8log2x
- log28x
- 3log2x
- Answer
-
- Incorrect. The exponent becomes a factor outside the logarithm. The correct answer is 8log2x.
- Correct. By the power property, log2x8=8log2x. You can’t simplify this further.
- Incorrect. The exponent becomes a factor outside the logarithm. The correct answer is 8log2x.
- Incorrect. You probably noticed that log28=3, so you used 3 instead of 8 when you pulled the exponent out to be a factor. However, the exponent must be pulled outside the logarithm to be a factor without any other changes. The correct answer is 8log2x.
Simplifying Logarithmic Expressions
The properties can be combined to simplify more complicated expressions involving logarithms.
Problem: Use the properties of logarithms to expand log10(abcd) into four simpler terms.
Answer
Use the quotient property to rewrite log10(abcd) as a difference of logarithms.
log10(abcd)=log10(ab)−log10(cd)
Now you have two logarithms, each with a product. Apply the product rule to each.
Be careful with the subtraction! Since all of log10cd is subtracted, you have to subtract both parts of the term: (log10c+log10d)
log10(ab)−log10(cd)=log10a+log10b−(log10c+log10d)
log10(abcd)=log10a+log10b−log10c−log10d
Problem: Simplify log6(ab)4, writing it as two separate terms.
Answer
Use the power property to rewrite log6(ab)4 as 4log6(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 log6(ab) as (log6a+log6b).
log6(ab)4=4log6(ab)=4(logba+log6b)
Use the distributive property.
log6(ab)4=4log6a+4log6b
Simplify log3x2y.
- 2(log3x+log3y)
- log3x2+log3y
- 2log3xy
- 2log3x+log3y
- Answer
-
- Incorrect. You may have started incorrectly by applying the power property, or you may have started correctly with the product property but then incorrectly applied the power property. The correct answer is 2log3x+log3y.
- Incorrect. While you correctly applied the product property first, log3x2 can be simplified further. The correct answer is 2log3x+log3y.
- Incorrect. You probably started incorrectly by applying the power property. Start with the product property. The correct answer is 2log3x+log3y.
- Correct. log3x2y=log3x2+log3y=2log3x+log3y
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 | bmbn=bm+n | logb(MN)=logbM+logbN |
Quotient Property | bmbn=bm−n | logb(MN)=logbM−logbN |
Power Property | (bm)n=bmn | logbMn=nlogbM |
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.