We are tasked with finding an equation that is equivalent to $8^5 = x$.

To convert this exponential form into a logarithmic form, we use the fact that:

$\log_b(a) = c \iff b^c = a$

Here, $b = 8$, $a = x$, and $c = 5$, so the equation $8^5 = x$ in logarithmic form is:

$\log_8(x) = 5$

Now, let's check the options:

• a) $\log_x 8 = 5$ → Incorrect (the base and argument are swapped).
• b) $\log_5 x = 8$ → Incorrect (wrong base and exponent).
• c) $\log_5 8 = x$ → Incorrect (wrong base and expression).
• d) $\log_8 x = 5$ → Correct.
• e) $\log_5 = x$ → Incorrect (no valid expression).
• f) $\log_x 5 = 8$ → Incorrect (wrong relationship).

The correct answer is d) $\log_8 x = 5$.

Would you like further details or explanations?

Here are 5 related questions:

1. How do you convert an equation from logarithmic form to exponential form?
2. What is the relationship between logarithms and exponents?
3. How do you solve logarithmic equations with different bases?
4. What are common properties of logarithms (like product and quotient rules)?
5. How can you simplify logarithmic expressions involving different bases?

Tip: Remember that the base in a logarithmic equation is the same as the base in the corresponding exponential form!