The equation in the image is:

$2 \log(3x + 2) = \log 121$

We need to find the value of $x$.

Step 1: Simplify the Equation

First, divide both sides of the equation by 2 to isolate the logarithm on the left-hand side:

$\log(3x + 2) = \frac{\log 121}{2}$

Step 2: Use Logarithm Properties

We can express $\frac{\log 121}{2}$ using the logarithm property $\log a^b = b \log a$:

$\log(3x + 2) = \log 121^{\frac{1}{2}} = \log \sqrt{121}$

Since $\sqrt{121} = 11$, the equation becomes:

$\log(3x + 2) = \log 11$

Step 3: Solve for $x$

Now, remove the logarithms by equating the arguments of the logarithms (because if $\log a = \log b$, then $a = b$):

$3x + 2 = 11$

Subtract 2 from both sides:

$3x = 9$

Finally, divide by 3:

$x = 3$

Answer:

The value of $x$ is $3$.

Would you like more details or have any questions?

Here are some related questions:

1. How do you solve logarithmic equations with different bases?
2. What are the properties of logarithms that are useful in solving equations?
3. Can you solve logarithmic equations using exponential form?
4. How do you verify if your solution for a logarithmic equation is correct?
5. What happens if the logarithm has a negative argument?

Tip: Always check if the argument inside the logarithm is positive since logarithms are only defined for positive real numbers.