The image contains a math exercise about logarithms and exponents. The task is to complete the table, which involves converting logarithmic forms into exponential forms and calculating the values. Here's a step-by-step explanation for the first few rows:

1. $\log_3 27 = a$

   • In exponential form: $3^a = 27$
   • Since $3^3 = 27$, we conclude that $a = 3$.

2. $\log_2 \frac{1}{64} = b$

   • In exponential form: $2^b = \frac{1}{64}$
   • Since $64 = 2^6$, we have $\frac{1}{64} = 2^{-6}$, so $b = -6$.

3. $\log_3 3 \sqrt{3} = c$

   • In exponential form: $3^c = 3 \sqrt{3}$
   • $3 \sqrt{3}$ can be written as $3^{3/2}$, so $c = \frac{3}{2}$.

4. $\log_4 8 = d$

   • In exponential form: $4^d = 8$
   • Using logarithms or estimation, $d \approx 1.5$ or $\frac{3}{2}$.

The other rows follow similar steps. Would you like further details on any of these steps or explanations for the remaining rows?

Related Questions:

1. How do you convert a logarithmic equation to its exponential form?
2. What properties of exponents are useful in solving logarithmic problems?
3. How do you handle fractional bases and roots in logarithmic equations?
4. What is the relationship between negative exponents and fractions?
5. How can logarithmic values be approximated when the base is not a whole number?

Tip: Always check if you can express a number as a power of the base when solving logarithmic equations.