To solve the equation [tex]\(\log_3(3x + 2) = \log_3(4x - 6)\)[/tex], we can use the property of logarithms that states if [tex]\(\log_b(A) = \log_b(B)\)[/tex], then [tex]\(A = B\)[/tex]. This property is valid because if the logarithms of two expressions are equal, then the expressions themselves must be equal.
Step-by-step solution:
1. Start with the given equation:
[tex]\[
\log_3(3x + 2) = \log_3(4x - 6)
\][/tex]
2. Use the property of logarithms mentioned above to set the arguments of the logarithms equal to each other:
[tex]\[
3x + 2 = 4x - 6
\][/tex]
3. Solve for [tex]\(x\)[/tex]. First, isolate the variable:
[tex]\[
3x + 2 = 4x - 6
\][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
2 = x - 6
\][/tex]
Add 6 to both sides:
[tex]\[
8 = x
\][/tex]
4. The solution to the equation is:
[tex]\[
x = 8
\][/tex]
So, the solution to the equation [tex]\(\log_3(3x + 2) = \log_3(4x - 6)\)[/tex] is [tex]\(x = 8\)[/tex].
Therefore, the correct answer is:
8
