Math  /  Algebra

QuestionQuestion 3 (4 points) Expand each logarithm on the left and match to its equivalent form on the right. Be sure to factor fully inside the logarithm first! \square log(3x3y)\log \left(\frac{3 x^{3}}{y}\right)
1. log(3)+log(x)+log(y)\log (3)+\log (x)+\log (y) \square log(3x2y2)\log \left(\frac{3}{x^{2}-y^{2}}\right)
2. log(3)+log(x)+3log(y)\log (3)+\log (x)+3 \log (y) log(x3y)\sim \log \left(\frac{x}{3 y}\right)
3. log(x)log(3)log(y)\log (x)-\log (3)-\log (y) log(3xy)\log (3 x y)
4. log(3)+3log(x)log(y)\log (3)+3 \log (x)-\log (y) \square log[3xy3]\log \left[3 x y^{3}\right] \square log(3(xy)3)\log \left(\frac{3}{(x-y)^{3}}\right)
5. log(3)log(x)log(y)\log (3) \log (x) \log (y)
6. log(x)log(3)log(y)\frac{\log (x)}{\log (3) \log (y)}
7. log(3)log(x)log(y)\frac{\log (3) \log (x)}{\log (y)} \square log(3x3y)\log (3 x-3 y)
8. log(3)2log(x)2log(y)\log (3)-2 \log (x)-2 \log (y)
9. log(3)+log(xy)\log (3)+\log (x-y)
10. log(3)log(xy)log(x+y)\log (3)-\log (x-y)-\log (x+y)
11. log(3)3log(xy)\log (3)-3 \log (x-y)

Studdy Solution

STEP 1

What is this asking? We need to rewrite each logarithm expression using logarithm properties to match it with a simplified version. Watch out! Remember the logarithm properties, like log(ab)=log(a)+log(b)\log(a \cdot b) = \log(a) + \log(b) and log(a/b)=log(a)log(b)\log(a/b) = \log(a) - \log(b), and don't mix them up!
Also, remember log(ab)=blog(a)\log(a^b) = b \cdot \log(a)!

STEP 2

1. Expand the first logarithm
2. Expand the second logarithm
3. Expand the third logarithm
4. Expand the fourth logarithm
5. Expand the fifth logarithm
6. Expand the sixth logarithm

STEP 3

We're looking at log(3x3y)\log \left(\frac{3 x^{3}}{y}\right).
Let's use the logarithm property log(a/b)=log(a)log(b)\log(a/b) = \log(a) - \log(b) to separate the numerator and denominator: log(3x3y)=log(3x3)log(y)\log \left(\frac{3 x^{3}}{y}\right) = \log(3x^3) - \log(y).

STEP 4

Now, we can use the property log(ab)=log(a)+log(b)\log(a \cdot b) = \log(a) + \log(b) to separate the terms in the numerator: log(3x3)log(y)=log(3)+log(x3)log(y)\log(3x^3) - \log(y) = \log(3) + \log(x^3) - \log(y).

STEP 5

Finally, we use the property log(ab)=blog(a)\log(a^b) = b \cdot \log(a) to bring down the exponent: log(3)+log(x3)log(y)=log(3)+3log(x)log(y)\log(3) + \log(x^3) - \log(y) = \log(3) + 3\log(x) - \log(y).
This matches option **4**.

STEP 6

The second expression is log(3x2y2)\log \left(\frac{3}{x^{2}-y^{2}}\right).
We can factor the denominator as a difference of squares: x2y2=(xy)(x+y)x^2 - y^2 = (x-y)(x+y).

STEP 7

So, we have log(3(xy)(x+y))\log \left(\frac{3}{(x-y)(x+y)}\right).
Using the division property gives us log(3)log((xy)(x+y))\log(3) - \log((x-y)(x+y)).

STEP 8

Using the multiplication property on the second term gives us log(3)(log(xy)+log(x+y))=log(3)log(xy)log(x+y)\log(3) - (\log(x-y) + \log(x+y)) = \log(3) - \log(x-y) - \log(x+y).
This matches option **10**.

STEP 9

We have log(x3y)\log \left(\frac{x}{3 y}\right).
Using the division property gives us log(x)log(3y)\log(x) - \log(3y).

STEP 10

Now, using the multiplication property gives us log(x)(log(3)+log(y))=log(x)log(3)log(y)\log(x) - (\log(3) + \log(y)) = \log(x) - \log(3) - \log(y).
This matches option **3**.

STEP 11

We have log(3xy)\log (3 x y).
Using the multiplication property twice gives us log(3)+log(x)+log(y)\log(3) + \log(x) + \log(y).
This matches option **1**.

STEP 12

We have log[3xy3]\log \left[3 x y^{3}\right].
Using the multiplication property gives us log(3)+log(x)+log(y3)\log(3) + \log(x) + \log(y^3).

STEP 13

Now, using the power property gives us log(3)+log(x)+3log(y)\log(3) + \log(x) + 3\log(y).
This matches option **2**.

STEP 14

We have log(3(xy)3)\log \left(\frac{3}{(x-y)^{3}}\right).
Using the division property gives us log(3)log((xy)3)\log(3) - \log((x-y)^3).

STEP 15

Now, using the power property gives us log(3)3log(xy)\log(3) - 3\log(x-y).
This matches option **11**.

STEP 16

1 matches with 4. 2 matches with 10. 3 matches with 3. 4 matches with 1. 5 matches with 2. 6 matches with 11.

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