Certainly! Let's expand the expression step-by-step, using logarithmic properties.
We are given the expression:
[tex]\[
\log_2\left(\frac{6 x^4}{y^7}\right)
\][/tex]
Using logarithm properties, we can break this expression into simpler parts. We'll make use of the following properties of logarithms:
1. [tex]\(\log_b(x \cdot y) = \log_b(x) + \log_b(y)\)[/tex]
2. [tex]\(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)[/tex]
3. [tex]\(\log_b(x^n) = n \cdot \log_b(x)\)[/tex]
Let's apply these properties step-by-step:
### Step 1: Breaking the fraction into separate logarithms
We start by using the property of logarithms that allows us to separate a fraction into a difference of logarithms:
[tex]\[
\log_2\left(\frac{6 x^4}{y^7}\right) = \log_2(6 x^4) - \log_2(y^7)
\][/tex]
### Step 2: Breaking the product [tex]\(6 x^4\)[/tex] into separate logarithms
Next, we'll separate the product inside the first logarithm using the property that allows us to add logarithms of products:
[tex]\[
\log_2(6 x^4) = \log_2(6) + \log_2(x^4)
\][/tex]
### Step 3: Expanding the exponent [tex]\(x^4\)[/tex]
Now we'll deal with the exponent inside the logarithm [tex]\(\log_2(x^4)\)[/tex]. Using the property that allows us to bring the exponent in front of the logarithm:
[tex]\[
\log_2(x^4) = 4 \log_2(x)
\][/tex]
### Step 4: Expanding the exponent [tex]\(y^7\)[/tex]
Similarly, we'll expand the exponent [tex]\(y^7\)[/tex] in the second logarithm:
[tex]\[
\log_2(y^7) = 7 \log_2(y)
\][/tex]
### Step 5: Combining all parts together
Replacing the expanded parts back into our expression, we get:
[tex]\[
\log_2\left(\frac{6 x^4}{y^7}\right) = \log_2(6) + \log_2(x^4) - \log_2(y^7)
\][/tex]
[tex]\[
= \log_2(6) + 4 \log_2(x) - 7 \log_2(y)
\][/tex]
So the expanded form of the given expression is:
[tex]\[
\log_2\left(\frac{6 x^4}{y^7}\right) = \log_2(6) + 4 \log_2(x) - 7 \log_2(y)
\][/tex]
Or, putting it in terms of [tex]\(\log_2[?]+\square \log_2 \square-\square \log_2 \square\)[/tex]:
[tex]\[
\log_2(6 x^4 / y^7) = \log_2(6) + 4 \log_2(x) - 7 \log_2(y)
\][/tex]
[tex]\[
= \log_2[6] + 4 \log_2[x] - 7 \log_2[y]
\][/tex]
