https://file.aiquickdraw.com/mathbot/file/1728513054465xqy8szp0.jpg

Let's simplify both parts of the question step by step.

### (a) \( 8^4 \cdot \log_8 3 \)

First, evaluate \( \log_8 3 \), which means "logarithm of 3 to the base 8." There's no simple integer value for this, so we leave it as is. Now focus on the given expression:

\[
8^4 \cdot \log_8 3
\]

Notice that \( 8^4 \) can be simplified:

\[
8^4 = (2^3)^4 = 2^{12}
\]

Thus, the expression becomes:

\[
2^{12} \cdot \log_8 3
\]

Without more simplification for \( \log_8 3 \), we would leave the final expression as:

\[
2^{12} \cdot \log_8 3
\]

### (b) \( \log_7 \left( \frac{1}{7^6} \right) \)

To simplify this:

\[
\log_7 \left( \frac{1}{7^6} \right) = \log_7 \left( 7^{-6} \right)
\]

Using the logarithmic property \( \log_b (a^x) = x \cdot \log_b a \), we can simplify:

\[
\log_7 \left( 7^{-6} \right) = -6 \cdot \log_7 7
\]

Since \( \log_7 7 = 1 \), the expression simplifies to:

\[
-6
\]

So, the answer for part (b) is \( -6 \).

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Let me know if you need more details or have any questions! Here are five related questions you might consider:

1. What is the exact value of \( \log_8 3 \)?
2. Can we express \( \log_8 3 \) using natural or common logarithms?
3. How would the answer change if the base in part (b) was different?
4. How does the property \( \log_b (a^x) = x \log_b a \

Simplify the following: (a) 8^4 * log_8 3, (b) log_7(1/7^6)

Simplify expressions involving logarithms and exponents. Problem (a) involves simplifying 8^4 * log_8 3 using properties of exponents. Problem (b) simplifies log_7(1/7^6) using logarithmic rules. Suitable for Grades 10-12.

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