Question

Solve the logarithmic equation log4(6−n)=log4(3n)

Answer 1

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 \( n \):

1. Add \( n \) to both sides:

\[
6 = 4n.
\]

2. 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:

1. For \( 6 - n \):

\[
6 - \frac{3}{2} = \frac{12}{2} - \frac{3}{2} = \frac{9}{2} > 0.
\]

2. 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

\[
\boxed{\frac{3}{2}}.
\]