Solve Logarithm Ratio: Finding log₄ₓ9/log₄ₓa Simplification

$\frac{\log_{4x}9}{\log_{4x}a}=$

To solve the given expression $\frac{\log_{4x}9}{\log_{4x}a}$ using the change-of-base formula, follow these steps:

• Step 1: Apply the change-of-base formula to both the numerator and the denominator expressions.
  This gives us: $\log_{4x}9 = \frac{\log_a 9}{\log_a (4x)}$ and $\log_{4x}a = \frac{\log_a a}{\log_a (4x)}$.
• Step 2: Substitute these into our original expression:
  $\frac{\log_{4x}9}{\log_{4x}a} = \frac{\frac{\log_a 9}{\log_a (4x)}}{\frac{\log_a a}{\log_a (4x)}}$.
• Step 3: Simplify the fraction:
  The $\log_a (4x)$ cancels out from the numerator and the denominator, leaving us with $\frac{\log_a 9}{\log_a a}$.
• Step 4: Further simplify using the fact that $\log_a a = 1$ because any number $a$ to the power of 1 is $a$.
  This results in $\frac{\log_a 9}{1} = \log_a 9$.

Therefore, the expression simplifies to $\log_a 9$.

The correct answer is $\log_a 9$, which matches choice 1.