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The problem you've provided is related to logarithms with non-standard bases, specifically with the bases written in subscript. Here's how we can solve the given questions:

### Given Information:
- \( ^3\log 2 = a \)
- \( ^2\log 7 = b \)

You are asked to find:

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

### Solution:

#### Part I: \( ^3\log 7 \)

To solve for \( ^3\log 7 \), we'll use the change of base formula. The general change of base formula is:

\[
^{c}\log d = \frac{\log d}{\log c}
\]

Using this, we need to express \( ^3\log 7 \) in terms of \( a \) and \( b \).

\[
^3\log 7 = \frac{\log 7}{\log 3}
\]

We are already given \( ^2\log 7 = b \), which means \( \log_2 7 = b \), but this doesn’t immediately help us in finding \( ^3\log 7 \). However, we might have to express this in terms of known logarithms or base conversions.

#### Part II: \( ^{27}\log 49 \)

For \( ^{27}\log 49 \), first, recognize that 49 is \( 7^2 \), so we can simplify:

\[
^{27}\log 49 = ^{27}\log (7^2) = 2 \times ^{27}\log 7
\]

Then, applying the change of base formula again:

\[
^{27}\log 7 = \frac{\log 7}{\log 27}
\]

Now, substituting this back:

\[
^{27}\log 49 = 2 \times \frac{\log 7}{\log 27}
\]

At this point, you would need the values of \( \log 7 \) and \( \log 27 \) to compute the exact result. Alternatively, express \( \log 27 \) in terms of smaller logarithms if additional information is given.

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Would you like more detailed steps for the computations? Here are five related questions that can help expand your understanding:

1. How does the change of base formula work in logarithms?
2. How can logarithmic identities simplify complex logarithmic expressions?
3. What are the properties of logarithms that deal with exponents, like \( \log (x^n) = n \log x \)?
4. How can logarithms with non-standard bases be converted to simpler base-10 or base-e logarithms?
5. Why is the change of base formula useful when working with different logarithmic bases?

**Tip:** When working with logarithms, the change of base formula is one of the most powerful tools, especially when you need to switch between uncommon logarithmic bases.

If ^3log 2 = a and ^2log 7 = b, find the values of I. ^3log 7 and II. ^27log 49.

Solve logarithmic problems using the change of base formula and logarithmic exponent rules. This problem explores finding values of ^3log 7 and ^27log 49 with given expressions, perfect for students in Grades 9-12.

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