Let's solve the logarithmic problems one by one from the image you uploaded:

a) ${}^5 \log 625 - {}^5 \log 25$

Using the logarithmic property:
$\log_b (m) - \log_b (n) = \log_b \left( \frac{m}{n} \right)$

This becomes: ${}^5 \log 625 - {}^5 \log 25 = {}^5 \log \left( \frac{625}{25} \right) = {}^5 \log 25$

Now, since ${}^5 \log 25 = 2$ (because $5^2 = 25$), we get:

$2 - 2 = 0$

Answer for (a): $0$

b) ${}^2 \log 4 + {}^2 \log 8$

Using the logarithmic property:
$\log_b (m) + \log_b (n) = \log_b (m \cdot n)$

This becomes: ${}^2 \log 4 + {}^2 \log 8 = {}^2 \log (4 \times 8) = {}^2 \log 32$

Now, we calculate ${}^2 \log 32$. Since $2^5 = 32$, we have:

${}^2 \log 32 = 5$

Answer for (b): $5$

c) ${}^2 \log 5 \times {}^5 \log 8$

We calculate each logarithm separately:

1. ${}^2 \log 5$ cannot be simplified easily, so we leave it as is.
2. ${}^5 \log 8$ — We express 8 as $2^3$, so: ${}^5 \log 8 = \frac{3}{\log_5 2} = \text{some value that involves approximation}$

Because one of these terms does not simplify easily in terms of elementary logarithms, this requires further approximation or analysis using logarithmic identities.

Would you like a detailed solution for part (c) or any other clarifications?

5 related questions:

1. How do you simplify logarithmic expressions involving different bases?
2. What are the key properties of logarithms?
3. How do we approach logarithms with irrational bases?
4. What methods exist to approximate complex logarithmic terms?
5. How do we convert logarithmic expressions to exponential form?

Tip: Always start by applying basic logarithmic properties such as product, quotient, or power rules to simplify expressions before attempting further steps.