Alright! Let's solve the given logarithmic equation step-by-step.
We start with the equation:
[tex]\[ 2 \log (3x + 2) = \log 121 \][/tex]
### Step 1: Simplify the Left Side
The left-hand side has a coefficient of 2. We can use the log property [tex]\(a \log b = \log(b^a)\)[/tex] to simplify:
[tex]\[ 2 \log (3x + 2) = \log ((3x + 2)^2) \][/tex]
So the equation becomes:
[tex]\[ \log ((3x + 2)^2) = \log 121 \][/tex]
### Step 2: Remove the Logarithms
Since the logs on both sides of the equation have the same base, we can set the arguments equal to each other:
[tex]\[ (3x + 2)^2 = 121 \][/tex]
### Step 3: Solve the Quadratic Equation
Next, we need to solve the quadratic equation:
[tex]\[ (3x + 2)^2 = 121 \][/tex]
Take the square root of both sides:
[tex]\[ 3x + 2 = \pm 11 \][/tex]
This gives us two possible equations to solve:
1. [tex]\( 3x + 2 = 11 \)[/tex]
2. [tex]\( 3x + 2 = -11 \)[/tex]
#### Solving [tex]\(3x + 2 = 11\)[/tex]:
Subtract 2 from both sides:
[tex]\[ 3x = 9 \][/tex]
Divide by 3:
[tex]\[ x = 3 \][/tex]
#### Solving [tex]\(3x + 2 = -11\)[/tex]:
Subtract 2 from both sides:
[tex]\[ 3x = -13 \][/tex]
Divide by 3:
[tex]\[ x = -\frac{13}{3} \][/tex]
### Step 4: Verify the Solutions
We need to verify whether these solutions are valid for the original logarithmic equation.
Let’s check [tex]\(x = 3\)[/tex]:
[tex]\[ \log(3(3) + 2) = \log 11 \][/tex]
[tex]\[ 2 \log 11 = \log 121 \][/tex]
This is correct, as [tex]\( \log 121 = 2 \log 11 \)[/tex].
Checking [tex]\(x = -\frac{13}{3}\)[/tex]:
[tex]\[ 3 \left(-\frac{13}{3}\right) + 2 = -13 + 2 = -11 \][/tex]
The logarithm of a negative number is undefined in the real number system. Therefore, [tex]\(x = -\frac{13}{3}\)[/tex] is not valid.
### Final Answer:
The value of [tex]\(x\)[/tex] is:
[tex]\[ x = 3 \][/tex]
