Sure, let's solve the expression [tex]\(\log_3 (x^3 y^2 z)\)[/tex] step-by-step using properties of logarithms.
### Step 1: Apply the property of logarithms
One of the properties of logarithms we can use is:
[tex]\[
\log_b (A \cdot B) = \log_b A + \log_b B
\][/tex]
Using this property, we can separate the terms inside the logarithm:
[tex]\[
\log_3 (x^3 y^2 z) = \log_3 (x^3) + \log_3 (y^2) + \log_3 (z)
\][/tex]
### Step 2: Simplify each term using another property of logarithms
Another property of logarithms we need to use is:
[tex]\[
\log_b (A^n) = n \cdot \log_b A
\][/tex]
Using this property, we can simplify each term:
[tex]\[
\log_3 (x^3) = 3 \cdot \log_3 (x)
\][/tex]
[tex]\[
\log_3 (y^2) = 2 \cdot \log_3 (y)
\][/tex]
[tex]\[
\log_3 (z) = \log_3 (z)
\][/tex]
### Step 3: Combine the simplified terms
Now that we have simplified each term, we combine them back together:
[tex]\[
\log_3 (x^3 y^2 z) = 3 \cdot \log_3 (x) + 2 \cdot \log_3 (y) + \log_3 (z)
\][/tex]
### Step 4: Express the result using the change of base formula
While the simplified expression using properties is correct, we can also express this in another form using the change of base formula:
[tex]\[
\log_b A = \frac{\log_k A}{\log_k b}
\][/tex]
In this case:
[tex]\[
\log_3 (x^3 y^2 z) = \frac{\log (x^3 y^2 z)}{\log (3)}
\][/tex]
By applying the property of logarithms [tex]\(\log_b (A \cdot B) = \log_b A + \log_b B\)[/tex] again, we separate out the terms inside the logarithm grouped by a common base (natural log or ln):
[tex]\[
\log_3 (x^3 y^2 z) = \frac{\log (x^3) + \log (y^2) + \log (z)}{\log (3)}
\][/tex]
And applying [tex]\(\log_b (A^n) = n \cdot \log_b A\)[/tex]:
[tex]\[
\log_3 (x^3 y^2 z) = \frac{3 \cdot \log (x) + 2 \cdot \log (y) + \log (z)}{\log (3)}
\][/tex]
So, the final result is:
[tex]\[
\log_3 (x^3 y^2 z) = \frac{3 \cdot \log (x) + 2 \cdot \log (y) + \log (z)}{\log (3)}
\][/tex]
This verifies that the final result of the given expression is indeed:
[tex]\[
\boxed{\frac{\log(x^3 y^2 z)}{\log(3)}}
\][/tex]
