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!