Let’s break down and solve each question step by step.

1. If $8^x = 3$, then $x =$

We need to solve for $x$ in the equation $8^x = 3$. To do this, we take the logarithm of both sides:

$\log(8^x) = \log(3)$

Using the logarithmic property $\log(a^b) = b \log(a)$, the equation becomes:

$x \log(8) = \log(3)$

Solve for $x$:

$x = \frac{\log(3)}{\log(8)}$

Thus, the correct answer is: D. $\frac{\log(3)}{\log(8)}$.

2. Which of the following is equivalent to $2^{x \log_2 a}$?

Start with the given expression:

$2^{x \log_2 a}$

Using the logarithmic property $a^{\log_b c} = c^{\log_b a}$, we can rewrite $2^{x \log_2 a}$ as:

$a^x$

Thus, the correct answer is: B. $a^x$.

Would you like a more detailed explanation of any part of this?

Related Questions:

1. How can logarithmic properties be used to simplify exponential equations like $8^x = 3$?
2. What are the steps to rewrite $a^{\log_b c}$ in terms of $c$ and $a$?
3. How can you verify $\log(a^b) = b \log(a)$ using examples?
4. Why is the base of the logarithm (e.g., base 10 or base 2) irrelevant when solving $\frac{\log(a)}{\log(b)}$?
5. How does the change-of-base formula for logarithms work in solving these problems?

Tip: Practice solving equations involving both logarithms and exponents to strengthen your understanding of their interrelation!