To find the value of [tex]\( x \)[/tex] for the given equation [tex]\( 2 \log (3x + 2) = \log 121 \)[/tex], let's break down the steps:
1. Simplify the logarithmic equation:
[tex]\[
2 \log (3x + 2) = \log 121
\][/tex]
We can use the logarithmic property [tex]\( a \log b = \log b^a \)[/tex]. Applying this property, we rewrite the equation:
[tex]\[
\log ( (3x + 2)^2 ) = \log 121
\][/tex]
2. Remove the logarithms:
Since the bases of the logarithms are the same and the equation is [tex]\( \log ((3x + 2)^2) = \log 121 \)[/tex], we can equate the arguments:
[tex]\[
(3x + 2)^2 = 121
\][/tex]
3. Solve the resulting quadratic equation:
To solve the quadratic equation, we first take the square root of both sides:
[tex]\[
3x + 2 = \pm \sqrt{121}
\][/tex]
Given that [tex]\( \sqrt{121} \)[/tex] is 11, we have:
[tex]\[
3x + 2 = 11 \quad \text{or} \quad 3x + 2 = -11
\][/tex]
4. Solve for x in each case:
For the first case:
[tex]\[
3x + 2 = 11
\][/tex]
Subtract 2 from both sides:
[tex]\[
3x = 9
\][/tex]
Divide both sides by 3:
[tex]\[
x = 3
\][/tex]
For the second case:
[tex]\[
3x + 2 = -11
\][/tex]
Subtract 2 from both sides:
[tex]\[
3x = -13
\][/tex]
Divide both sides by 3:
[tex]\[
x = -\frac{13}{3}
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] that satisfy the original equation are:
[tex]\[
x = 3 \quad \text{and} \quad x = -\frac{13}{3}
\][/tex]
