Properties of Logarithmic Functions
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 logarithmsA calculation in which the exponent `y` in `x = b^y` is found when given `x` and `b`; the corresponding notation is `log_b x = y`. . 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 exponentWhen a number is expressed in the form `a^b`, `b` is the exponent. The exponent indicates how many times the base is used as a factor. Power and exponent mean the same thing. properties.
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Properties of Exponents
Product of powers: `b^mb^n=b^(m+n)` Quotient of powers: `b^m/b^n=b^(m-n)` Power of a power: `(b^m)^n=b^(mn)`
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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 `log_3 3^2=y`. Remember `log_bx=y hArrb^y=x`, so `log_3 3^2=y` means `3^y=3^2` and `y` must be `2`, which means `log_3 3^2=2`. You will get the same answer that `log_3 3^2` 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 baseThe expression that is being raised to a power when using exponential notation. In `5^3`, `5` is the base which is the number that is repeatedly multiplied. `5^3 = 5 * 5 * 5`. In `a^b`, the base is `a`. , you add the exponents. With logarithms, the logarithm of a product is the sum of the logarithms.
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Logarithm of a Product
The logarithm of a product is the sum of the logarithms:
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Let’s try the following example.
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Example |
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Problem |
Use the product property to rewrite `log_2 (4⋅8)`. |
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`log_2(4*8)=log_2 4+log_2 8`
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Use the product property to write as a sum. |
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`log_2 4+log_2 8=` `log_2 2^2+log_2 2^3=` `2+3=5` |
Simplify each addend, if possible. In this case, you can simplify both addends. First rewrite `log_2 4` as `log_2 2^2` and `log_2 8` as `log_2 2^3`, and 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. |
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Answer |
`log_2(4*8)=5` |
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Another way to simplify `log_2 (4*8)` would be to multiply `4` and `8` as a first step.
`log_2 (4*8)=log_2 32=5` because `2^5=32`
You get the same answer `log_2(4*8)=5` as in the example!
Notice the similarity to the exponent property: `b^mb^n=b^(m+n)`, while `log_b (MN)=log_bM+log_bN`. In both cases, a product becomes a sum.
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Example |
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Problem |
Use the product property to rewrite `log_3(9x)`. |
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`log_3(9x)=log_3 9 + log_3 x` |
Use the product property to write as a sum. |
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`log_3 9 + log_3 x=log_3 3^2+ log_3 x=2+log_3x` |
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` and 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. |
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Answer |
`log_3(9x) = 2 + log_3 x` |
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If the product has many factors, you just add the individual logarithms:
`log_b(ABCD)=log_bA+log_bB+log_bC+log_bD`
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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`
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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.
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Logarithm of a Quotient `log_b (M/N)=log_b M-log_b N`
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With both properties: `b^m/b^n=b^(m-n)` and `log_b(M/N)=log_b M-log_b N`, a quotient becomes a difference.
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Example |
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Problem |
Use the quotient property to rewrite `log_x(x/2)`. |
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`log_2(x/2)=log_2 x-log_2 2` |
Use the quotient property to rewrite as a difference. |
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Answer |
`log_2(x/2) =log_2x - 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`. |
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Which of these is equivalent to `log_3(81/a)`?
A) `4 - log_3 a`
B) `4/log_3 a`
C) `log_3 (4-a)`
D) `log_3(4/a)`
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The remaining exponent property was power of a power: `(b^m)^n=b^(mn)`. The similarity with the logarithm of a power is a little harder to see.
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Logarithm of a Power `log_b M^n=n log_bM`
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With both properties, `(b^m)^n=b^(mn)` and `log_b M^n=n log_bM`, the power “`n`” becomes a factor.
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Example |
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Problem |
Use the power property to simplify `log_3 9^4`. |
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`log_3 9^4=4log_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`. |
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`4 log_3 9 = 4*2` |
You may be able to recognize by now that since `3^2 = 9`, `log_3 9 = 2`. |
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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`!
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Example |
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Problem |
Use the properties of logarithms to rewrite `log_4 64^x`. |
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`log_4 64^x = x log_4 64` `=x log_4 4^3` `=x*3` |
Use the power property to rewrite `log_4 64^x` as `x log_4 64`.
`64=4*4*4=4^3` Rewrite `log_4 64` as `log_4 4^3`, then use the property `log_b b^x=x` to simplify `log_4 4^3`. Or, you may be able to recognize by now that since `4^3 = 64`, `log_4 64 = 3`. |
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Answer |
`log_4 64^x=3x` |
Multiply the factors. |
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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`
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The properties can be combined to simplify more complicated expressions involving logarithms.
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Example |
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Problem |
Use the properties of logarithms to expand `log_10 ((ab)/(cd))` into four simpler terms. |
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`log_10 ((ab)/(cd))=log_10(ab)-log_10(cd)` |
Use the quotient property to rewrite `log_10 ((ab)/(cd))` as a difference of logarithms. |
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`log_10(ab)-log_10(cd)=log_10 a+log_10 b-(log_10 c + log_10 d)` |
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)` |
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Answer |
`log_10 ((ab)/(cd))=log_10 a+log_10 b-log_10 c-log_10 d` |
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Example |
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Problem |
Simplify `log_6(ab)^4`, writing it as two separate terms. |
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`log_6(ab)^4=4 log_6 (ab)` `=4(log_b a + log_6 b)` |
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)`. |
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Answer |
`log_6(ab)^4=4 log_6 a + 4 log_6 b` |
Use the distributive property. |
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Simplify `log_3 x^2y`.
A) `2(log_3x + log_3y)`
B) `log_3 x^2 + log_3y`
C) `2 log_3 xy`
D) `2 log_3x + log_3 y`
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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.
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Exponents |
Logarithms |
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Product Property |
`b^mb^n=b^(m+n)` |
`log_b(MN)=log_bM+log_bN` |
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Quotient Property |
`b^m/b^n=b^(m-n)` |
`log_b(M/N)=log_bM-log_bN` |
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Power Property |
`(b^m)^n=b^(mn)` |
`log_bM^n=n log_b M` |
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.