https://file.aiquickdraw.com/mathbot/file/17293149202366ke471yj.jpg

Let's analyze and solve the two logarithmic expressions given in the image:

### (a) \(\log_8 \left(\frac{1}{8}\right)\)
This is a logarithmic expression in base 8, and we need to find the value of:

\[
\log_8 \left(\frac{1}{8}\right) = x
\]

By the definition of logarithms, this means:

\[
8^x = \frac{1}{8}
\]

We know that \(\frac{1}{8} = 8^{-1}\), so we can rewrite the equation as:

\[
8^x = 8^{-1}
\]

Since the bases are the same, we can equate the exponents:

\[
x = -1
\]

Thus, \(\log_8 \left(\frac{1}{8}\right) = -1\).

### (b) \(\log 1000\)
Here, the expression is using a logarithm with no explicit base, meaning it's a common logarithm with base 10:

\[
\log 1000 = x
\]

This means:

\[
10^x = 1000
\]

Since \(1000 = 10^3\), we have:

\[
10^x = 10^3
\]

Equating the exponents:

\[
x = 3
\]

Thus, \(\log 1000 = 3\).

### Final Answers:
- (a) \(\log_8 \left(\frac{1}{8}\right) = -1\)
- (b) \(\log 1000 = 3\)

Would you like a more detailed explanation or have any further questions?

Here are some related questions you might find interesting:
1. How can we express \(\log_8(64)\) in terms of a simple number?
2. What is the general relationship between logarithms with different bases?
3. Can you solve \(\log_2(32)\) without using a calculator?
4. How do we simplify expressions like \(\log_3(27)\)?
5. What happens when we solve logarithmic expressions with decimal numbers, like \(\log(0.01)\)?

**Tip:** Remember that logarithms are exponents. The equation \(\log_b(x) = y\) means that \(b^y = x\), so logarithmic problems can often be solved by equating powers!

Evaluate the logarithmic expressions: (a) log_8(1/8) and (b) log 1000.

Learn how to evaluate logarithmic expressions such as log_8(1/8) and log 1000. This step-by-step solution covers key concepts in logarithms and exponents. Suitable for students in Grades 9-11.

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