Let's start by understanding the relationship between exponential equations and logarithmic equations.
The general rule for converting an exponential equation to a logarithmic equation is as follows:
- If you have an exponential equation of the form [tex]\(a^b = c\)[/tex], you can convert it into the logarithmic form as [tex]\(\log_a(c) = b\)[/tex].
Given the exponential equation [tex]\(6^x = 216\)[/tex]:
1. Identify the base [tex]\(a\)[/tex], exponent [tex]\(b\)[/tex], and the result [tex]\(c\)[/tex]. Here, [tex]\(a = 6\)[/tex], [tex]\(b = x\)[/tex], and [tex]\(c = 216\)[/tex].
2. Apply the rule for converting exponential equations to logarithmic equations. According to the rule, we can write:
[tex]\[
\log_6 (216) = x
\][/tex]
This logarithmic equation [tex]\(\log_6 (216) = x\)[/tex] represents the relationship in terms of logarithms.
Therefore, the correct logarithmic equation equivalent to the given exponential equation [tex]\(6^x = 216\)[/tex] is:
[tex]\[
\log_6 (216) = x
\][/tex]
This corresponds to option C. So, the correct answer is:
C. [tex]\(\log_6 (216) = x\)[/tex]
