Given the equations:

$\log_{10}(ab) = \log_{10}(b) = c \quad \text{and} \quad \log_{10}(100) = c$

Let's analyze them step by step:

1. Interpreting the equations:

   • $\log_{10}(100) = c$ implies $c = 2$, because $10^2 = 100$.
   • $\log_{10}(ab) = \log_{10}(b) = c$ implies: $\log_{10}(ab) = c = 2 \quad \text{and} \quad \log_{10}(b) = 2$
   • If $\log_{10}(b) = 2$, then $b = 100$.
   • If $\log_{10}(ab) = 2$, then $ab = 10^2 = 100$.

2. From $ab = 100$ and $b = 100$:

   $a \times 100 = 100 \implies a = 1$

Thus, $a = 1$ and $b = 100$.

3. Evaluating the options:

   • Option 1: $452 + 852 > a + b$

     Substituting $a = 1$ and $b = 100$: $452 + 852 = 1304 \quad \text{and} \quad a + b = 1 + 100 = 101$ Clearly, $1304 > 101$. So, this statement might be true.

   • Option 2: $5ck = a^{100}$ for some integer value of $k$

     Substituting $c = 2$ and $a = 1$: $5 \times 2 \times k = 1^{100} = 1$ However, $5 \times 2 \times k = 10k$. For $k$ to be an integer, $10k = 1$ is not possible. So, this statement cannot be true.

   • Option 3: $(452)(852) > ab$

     Substituting $ab = 100$: $(452)(852) = 385104 \quad \text{and} \quad ab = 100$ Clearly, $385104 > 100$. So, this statement might be true.

   • Option 4: None of the above

     Since two of the statements might be true, this option is incorrect.

Conclusion:

Both Option 1 and Option 3 might be true.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. What are the properties of logarithms?
2. How can logarithmic equations be used to solve exponential equations?
3. What are the key differences between natural logarithms and common logarithms?
4. How would the solution change if the base of the logarithm was different?
5. How can logarithms be applied in solving real-world problems?

Tip: Remember that logarithms are the inverses of exponentiation. This relationship is crucial when solving logarithmic equations.