Question

Solve the logarithmic equation log(3x−5)=log(2x−3)
(1 point)
x=

Answer 1

To solve the equation \( \log(3x - 5) = \log(2x - 3) \), we can use the property that if \( \log(a) = \log(b) \), then \( a = b \), provided that both \( a \) and \( b \) are positive.

Starting from the equation:

\[
3x - 5 = 2x - 3
\]

Now, we can isolate \( x \):

1. Subtract \( 2x \) from both sides:

\[
3x - 2x - 5 = -3
\]

2. This simplifies to:

\[
x - 5 = -3
\]

3. Now, add 5 to both sides:

\[
x = 2
\]

Next, we need to check if this solution is valid by substituting \( x = 2 \) back into the logarithmic expressions to ensure they are positive:

\[
3(2) - 5 = 6 - 5 = 1 \quad \text{(positive)}
\]
\[
2(2) - 3 = 4 - 3 = 1 \quad \text{(positive)}
\]

Since both arguments of the logarithms are positive, the solution is valid.

Thus, the final solution is:

\[
\boxed{2}
\]