To solve the expression [tex]\(\log_6(6x)\)[/tex], we rely on the properties of logarithms. Here is the step-by-step solution:
1. Identify the expression:
[tex]\[
\log_6(6x)
\][/tex]
2. Use the property of logarithms that allows the argument to be split into a product:
The property [tex]\(\log_b(mn) = \log_b(m) + \log_b(n)\)[/tex] states that the logarithm of a product is the sum of the logarithms. Apply this property:
[tex]\[
\log_6(6x) = \log_6(6) + \log_6(x)
\][/tex]
3. Simplify [tex]\(\log_6(6)\)[/tex]:
[tex]\(\log_6(6)\)[/tex] is a special case, because any logarithm where the base and the argument are the same is 1. Therefore:
[tex]\[
\log_6(6) = 1
\][/tex]
4. Combine the results:
Substituting back into the expression, we have:
[tex]\[
\log_6(6x) = 1 + \log_6(x)
\][/tex]
Therefore, the fully simplified expression is:
[tex]\[
\boxed{1 + \log_6(x)}
\][/tex]
