To solve the logarithmic equation
\[ \log_4(6 - n) = \log_4(3n), \]
we can set the arguments equal to each other, since the logarithm function is one-to-one. Thus, we have:
\[ 6 - n = 3n. \]
Now, we will solve for \( n \):
- Add \( n \) to both sides:
\[ 6 = 4n. \]
- Divide both sides by 4:
\[ n = \frac{6}{4} = \frac{3}{2}. \]
Next, we need to check if this value of \( n \) is valid in the context of the logarithm. The arguments of the logarithms must be positive:
- For \( 6 - n \):
\[ 6 - \frac{3}{2} = \frac{12}{2} - \frac{3}{2} = \frac{9}{2} > 0. \]
- For \( 3n \):
\[ 3 \cdot \frac{3}{2} = \frac{9}{2} > 0. \]
Since both arguments are positive, \( n = \frac{3}{2} \) is a valid solution.
Thus, the solution to the equation is
\[ \boxed{\frac{3}{2}}. \]
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