To expand the logarithmic expression [tex]\(\log_7(6st)\)[/tex] using the Laws of Logarithms, follow these steps:
### Step-by-Step Solution:
1. Understand the Product Rule for Logarithms:
The product rule for logarithms states that:
[tex]\[
\log_b(m \cdot n) = \log_b(m) + \log_b(n)
\][/tex]
This rule applies when you have a product inside the logarithm and allows you to separate it into the sum of individual logarithms.
2. Identify the parts inside the logarithm:
The expression inside the logarithm is [tex]\(6st\)[/tex]. This is a product of three terms: 6, [tex]\(s\)[/tex], and [tex]\(t\)[/tex].
3. Apply the Product Rule iteratively:
First, separate the logarithm of the product of 6 and [tex]\(st\)[/tex]:
[tex]\[
\log_7(6st) = \log_7(6) + \log_7(st)
\][/tex]
Next, apply the product rule again to the [tex]\(\log_7(st)\)[/tex] term:
[tex]\[
\log_7(st) = \log_7(s) + \log_7(t)
\][/tex]
4. Combine all the separated terms:
Putting it all together, we have:
[tex]\[
\log_7(6st) = \log_7(6) + \log_7(s) + \log_7(t)
\][/tex]
### Final Expanded Expression:
[tex]\[
\log_7(6) + \log_7(s) + \log_7(t)
\][/tex]
Thus, the expanded form of the logarithmic expression [tex]\(\log_7(6st)\)[/tex] is:
[tex]\[
\log_7(6) + \log_7(s) + \log_7(t)
\][/tex]
