https://file.aiquickdraw.com/mathbot/file/1726105629638y55ps1wp.jpg

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.

If 3log 2 = a and 2log 7 = b, find: 1. 3log 7, 2. 27log 49

Explore how to solve logarithmic expressions such as 3log7 and 27log49 using change of base and logarithmic rules. This step-by-step guide is designed for students in Grades 11-12 and covers key logarithmic concepts.

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