Question

Solve: log3 (2x-1) = log3 (3x)

Answer 1

To solve the equation \( \log_3 (2x - 1) = \log_3 (3x) \), we can use the property of logarithms that states if \( \log_b(a) = \log_b(c) \), then \( a = c \), provided that \( a > 0 \) and \( c > 0 \).

Applying this property to our equation:

\[
2x - 1 = 3x
\]

Now, let's solve for \( x \):

\[
2x - 1 = 3x
\]

Subtract \( 2x \) from both sides:

\[
-1 = 3x - 2x
\]

This simplifies to:

\[
-1 = x
\]

Now, we need to check if this value satisfies the conditions for the logarithm's argument to be positive. Substitute \( x = -1 \) back into the arguments of the logarithms:

1. For the left side:
\[
2(-1) - 1 = -2 - 1 = -3
\]
(not valid since logarithm of a negative number is undefined)

2. For the right side:
\[
3(-1) = -3
\]
(not valid since logarithm of a negative number is undefined)

Since neither \( 2x - 1 \) nor \( 3x \) is positive when \( x = -1 \), there are no valid solutions for this equation.

Thus, the conclusion is that there are **no solutions** for the equation \( \log_3 (2x - 1) = \log_3 (3x) \).