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What is the solution of [tex]\log _3(3x + 2) = \log _3(4x - 6)[/tex]?

A. [tex]\(-8\)[/tex]
B. [tex]\(-6\)[/tex]
C. [tex]\(8\)[/tex]
D. [tex]\(9\)[/tex]

Sagot :

GinnyAnsweridn
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
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