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Which logarithmic equation is equivalent to this exponential equation?

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

A. [tex]\(\log_6 x = 216\)[/tex]
B. [tex]\(\log_x 216 = x\)[/tex]
C. [tex]\(\log_x 6 = 216\)[/tex]
D. [tex]\(\log_6 216 = x\)[/tex]

Sagot :

GinnyAnsweridn
Let's analyze the given exponential equation and convert it into its equivalent logarithmic form.

The given equation is:
[tex]\[ 6^x = 216 \][/tex]

To convert an exponential equation [tex]\( a^b = c \)[/tex] into logarithmic form, it can be written as [tex]\( \log_a(c) = b \)[/tex].

In our case:
- The base [tex]\( a \)[/tex] is 6.
- The exponent [tex]\( b \)[/tex] is [tex]\( x \)[/tex].
- The result [tex]\( c \)[/tex] is 216.

Applying the logarithmic transformation:
[tex]\[ \log_6(216) = x \][/tex]

Now, let's compare this with the given answer choices:

A. [tex]\( \log_6 x = 216 \)[/tex] - This incorrectly suggests that [tex]\( x \)[/tex] is the number being log-transformed and equals 216.

B. [tex]\( \log_x 216 = x \)[/tex] - This incorrectly uses [tex]\( x \)[/tex] as the base of the logarithm.

C. [tex]\( \log_x 6 = 216 \)[/tex] - This incorrectly places [tex]\( x \)[/tex] as the base.

D. [tex]\( \log_6 216 = x \)[/tex] - This correctly represents the original exponential equation in logarithmic form.

So, the correct answer is:
[tex]\[ \boxed{D. \log_6 216 = x} \][/tex]
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