Let's solve the logarithmic problems presented in the image:

1. a. $\log_2 32$

   To solve $\log_2 32$, we need to determine the power to which 2 must be raised to get 32: $2^5 = 32$ Hence, $\log_2 32 = 5$.

2. b. $\log_7 \left(\frac{1}{343}\right)$

   We know that $343 = 7^3$. Since the logarithm involves a reciprocal, we can apply the property: $\log_b \left(\frac{1}{x}\right) = -\log_b(x)$ Therefore, $\log_7 \left(\frac{1}{343}\right) = -\log_7(343) = -3$ because $7^3 = 343$.

3. c. $\log 10000$

   Since no base is specified, we assume this is the common logarithm (base 10). We know that: $10^4 = 10000$ Hence, $\log 10000 = 4$.

Let me know if you need more details on any step!

Relative questions:

1. How do you calculate logarithms with different bases?
2. What are the key properties of logarithms?
3. How can you simplify expressions involving $\log_b (1/x)$?
4. What is the difference between common logarithms and natural logarithms?
5. How do you solve logarithmic equations involving multiple terms?

Tip: Remember that the base of a logarithm determines how fast the function grows or shrinks. For example, $\log_2(x)$ grows faster than $\log_3(x)$.