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Which logarithmic equation is equivalent to this exponential equation?

[tex]\[ 6^x = 216 \][/tex]

A. [tex]\(\log_x 216 = x\)[/tex]

B. [tex]\(\log_x 6 = 216\)[/tex]

C. [tex]\(\log_6 216 = x\)[/tex]

D. [tex]\(\log_6 x = 216\)[/tex]

Sagot :

GinnyAnswer
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]
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