To rewrite [tex]\(\log_7(4x) + \log_7(3y)\)[/tex] as a single logarithm, we can utilize the properties of logarithms. The property that is particularly useful here is the product property of logarithms, which states:
[tex]\[
\log_b(a) + \log_b(c) = \log_b(a \cdot c)
\][/tex]
Given this, let's apply the property step by step:
1. Identify the given logarithmic expression:
[tex]\[
\log_7(4x) + \log_7(3y)
\][/tex]
2. Use the product property to combine the logarithms:
[tex]\[
\log_7(4x) + \log_7(3y) = \log_7((4x) \cdot (3y))
\][/tex]
3. Simplify the expression inside the logarithm:
[tex]\[
(4x) \cdot (3y) = 4 \cdot 3 \cdot x \cdot y = 12xy
\][/tex]
4. Substitute back into the logarithm:
[tex]\[
\log_7((4x) \cdot (3y)) = \log_7(12xy)
\][/tex]
Therefore, the expression [tex]\(\log_7(4x) + \log_7(3y)\)[/tex] can be rewritten as a single logarithm:
[tex]\[
\log_7(12xy)
\][/tex]
