Answer :
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]
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]
