Sure! Let's solve the problem step-by-step using the Product Rule of Logarithms.
The Product Rule of Logarithms states that for any positive numbers [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ \ln(x \cdot y) = \ln(x) + \ln(y) \][/tex]
We need to apply this rule to the expression [tex]\( \ln(6a + 9b) \)[/tex].
First, let's factor the expression inside the logarithm:
[tex]\[ 6a + 9b \][/tex]
Notice that we can factor out a 3 from both terms:
[tex]\[ 6a + 9b = 3(2a + 3b) \][/tex]
Now, we have:
[tex]\[ \ln(6a + 9b) = \ln(3(2a + 3b)) \][/tex]
According to the Product Rule of Logarithms, we can separate the logarithm of the product into the sum of logarithms:
[tex]\[ \ln(3(2a + 3b)) = \ln(3) + \ln(2a + 3b) \][/tex]
So, the completely expanded expression equivalent to [tex]\( \ln(6a + 9b) \)[/tex] is:
[tex]\[ \boxed{\text{In}(6a + 9b) = \text{In}(3) + \text{In}(2a + 3b)} \][/tex]
