Let's analyze the problem in the image:

Given:

• $^3\log 2 = a$
• $^2\log 7 = b$

We need to find:

1. $^3\log 7$
2. $^{27}\log 49$

Step 1: Find $^3\log 7$

Using the change of base formula: $^a\log b = \frac{\log b}{\log a}$ We can express $^3\log 7$ in terms of the given information. Since $^2\log 7 = b$, and by the change of base property for logarithms:

$^3\log 7 = \frac{^2\log 7}{^2\log 3}$

But since $^2\log 7 = b$, we still need $^2\log 3$. However, this is not provided, so this requires estimation or assumption based on logarithmic identities. Let's assume the problem simplifies with values of $a$ and $b$.

Step 2: Find $^{27}\log 49$

We know that 49 is $7^2$, so: $^{27}\log 49 = ^{27}\log (7^2)$

Using the logarithmic power rule, $^a\log (b^n) = n \cdot ^a\log b$: $^{27}\log 49 = 2 \cdot ^{27}\log 7$

Now, we use the change of base formula again, simplifying: $^{27}\log 7 = \frac{^2\log 7}{^2\log 27}$ Given $^2\log 7 = b$, and if we have or estimate $^2\log 27$, the problem becomes straightforward.

Would you like further assistance or clarification on these steps? Here are some related questions:

1. How can we estimate $^2\log 3$ to solve $^3\log 7$?
2. What is the logarithmic change of base formula and how does it apply here?
3. How can we simplify logarithms with large bases, such as $^{27}\log 49$?
4. What additional information is needed to complete this problem definitively?
5. How can powers and roots simplify logarithmic expressions like $^{27}\log 49$?

Tip: Always break down complex logarithmic expressions using basic properties such as the power rule or the change of base formula for simplification.