Let's break down the problem step by step.

Step 1: Rewrite the logarithmic equation in its exponential form.

The given logarithmic equation is:

$\log_2 (x - 5) = 3$

The general rule to convert a logarithmic equation $\log_b (y) = c$ into exponential form is:

$b^c = y$

So, applying this rule to the given equation:

$\log_2 (x - 5) = 3 \quad \Rightarrow \quad 2^3 = x - 5$

Thus, the equivalent exponential form is:

$2^3 = x - 5$

Step 2: Solve for $x$.

From the exponential form, we have:

$2^3 = x - 5$

Simplifying the exponent:

$8 = x - 5$

Now, solve for $x$ by adding 5 to both sides:

$x = 8 + 5$

$x = 13$

Final Answer:

• The equivalent exponential form is $2^3 = x - 5$.
• The value of $x$ is $13$.

Would you like further details on any step?

Here are some related questions you might consider:

1. What is the general formula to convert logarithmic expressions to exponential form?
2. How do you handle logarithms with different bases?
3. How would the process change if the logarithmic equation had a negative exponent?
4. Can logarithms of negative numbers be calculated?
5. How is solving logarithmic equations applied in real-world problems?

Tip: Always verify if the result satisfies the domain restrictions of the logarithmic function. In this case, $x - 5 > 0$, which ensures $x > 5$, and the solution $x = 13$ is valid.