Here’s the solution to the questions from the uploaded image:

Question 9: Evaluate the logarithmic expressions

a) $\log_2 4$:
$\log_2 4 = x \quad \Rightarrow \quad 2^x = 4.$
Since $2^2 = 4$, the solution is: $\log_2 4 = 2.$

b) $\log_3 27$:
$\log_3 27 = x \quad \Rightarrow \quad 3^x = 27.$
Since $3^3 = 27$, the solution is: $\log_3 27 = 3.$

c) $\log_4 64$:
$\log_4 64 = x \quad \Rightarrow \quad 4^x = 64.$
Since $4^3 = 64$, the solution is: $\log_4 64 = 3.$

d) $\log_5 1$:
$\log_5 1 = x \quad \Rightarrow \quad 5^x = 1.$
Since $5^0 = 1$, the solution is: $\log_5 1 = 0.$

e) $\log_2 \left(\frac{1}{2}\right)$:
$\log_2 \left(\frac{1}{2}\right) = x \quad \Rightarrow \quad 2^x = \frac{1}{2}.$
Since $2^{-1} = \frac{1}{2}$, the solution is: $\log_2 \left(\frac{1}{2}\right) = -1.$

f) $\log_3 \sqrt{3}$:
$\log_3 \sqrt{3} = x \quad \Rightarrow \quad 3^x = \sqrt{3}.$
Since $\sqrt{3} = 3^{1/2}$, the solution is: $\log_3 \sqrt{3} = \frac{1}{2}.$

Question 10: Why can $\log_3(-9)$ not be evaluated?

The logarithm of a negative number is undefined in the real number system because the base raised to any real power cannot result in a negative value.
For example:
$3^x \neq -9 \quad \text{for any real } x.$
Thus, $\log_3(-9)$ is undefined.

Question 11: Find coordinates with $y$-values $-2, -1, 0, 1, 2$ for each function

a) $y = \log_2 x$:

To find $x$ values for $y = -2, -1, 0, 1, 2$:
$x = 2^y.$

b) $y = \log_{10} x$:

To find $x$ values for $y = -2, -1, 0, 1, 2$:
$x = 10^y.$

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Expanding Questions:

1. What is the difference between real and complex logarithms?
2. How do logarithmic functions behave asymptotically as $x \to \infty$?
3. What are the properties of logarithmic scales, e.g., in sound intensity or pH levels?
4. How can logarithmic functions be used to solve exponential equations?
5. What is the significance of the change-of-base formula in logarithms?

Tip:

Logarithms are the inverse of exponential functions. Understanding their relationship helps in solving problems that involve both functions effectively.