The equation in the image appears to be:

$\log_7(x^2 + 1, 52) = 0$

This is a logarithmic equation, and we can solve for $x$ by applying the properties of logarithms. To solve it:

1. Use the definition of logarithms: $\log_b(a) = c \quad \text{implies} \quad b^c = a$ In this case, it becomes: $7^0 = x^2 + 1, 52$

2. Simplify: Since $7^0 = 1$, we have: $x^2 + 1 = 1$

3. Solve for $x$: $x^2 = 0$ $x = 0$

So the solution is $x = 0$.

Would you like me to explain any steps further or do you have any questions about this solution? Here are 5 related questions to explore:

1. What are the key properties of logarithms?
2. How do you solve logarithmic equations with different bases?
3. What is the change of base formula in logarithms?
4. How can you use logarithms to solve exponential equations?
5. Can this equation have any other solutions if we change the base or the expression?

Tip: Always ensure the argument of a logarithm is positive, as the logarithm of a non-positive number is undefined in real numbers.