Question

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

Answer 1

To solve the logarithmic equation \( \log(3x - 5) = \log(2x - 3) \), we can use the property that if the logarithms of two expressions are equal, then the expressions themselves are equal (given that both expressions are positive).

So we set the arguments of the logarithms equal to each other:

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

Now, solve for \( x \):

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

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

This simplifies to:

\[
x - 5 = -3
\]

2. Add 5 to both sides:

\[
x = 2
\]

Now, we need to check if this solution is valid by plugging it back into the original logarithmic expressions to ensure they are defined (i.e., positive):

- For \( x = 2 \):

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

Since both expressions are positive, the solution is valid.

Thus, the solution to the equation is:

\[
\boxed{2}
\]