Question

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

Answer 1

To solve the logarithmic equation \( \log_4(6 - n) = \log_4(3n) \), we can use the property of logarithms that states if \(\log_b(x) = \log_b(y)\), then \(x = y\).

Applying this property to our equation, we have:

\[
6 - n = 3n
\]

Next, we can solve for \(n\). First, let's add \(n\) to both sides:

\[
6 = 3n + n
\]

This simplifies to:

\[
6 = 4n
\]

Now, divide both sides by 4:

\[
n = \frac{6}{4} = \frac{3}{2}
\]

So, the solution is:

\[
n = \frac{3}{2}
\]

We should also verify that this solution satisfies the conditions of the original logarithmic equation. We need to check that the arguments of the logarithms are positive when \( n = \frac{3}{2} \):

1. \(6 - n = 6 - \frac{3}{2} = \frac{12}{2} - \frac{3}{2} = \frac{9}{2} > 0\)
2. \(3n = 3 \cdot \frac{3}{2} = \frac{9}{2} > 0\)

Since both arguments are positive, the solution is valid.

Thus, the final answer is:

\[
n = \frac{3}{2}
\]