If log4 (x) = 12 , what is log2 (x/4) ?

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Given that log(4) x = 12

We need to find log(2) x/4

First we will use logarithm properties to simplify.

We know that:

log a/b = log a - log b

==> log2 (x/4) = log2 x - log2 4

Now we will rewrite 4= 2^2

==> log2 (x/4) = log2 x - log2 2^2

Now we know that log a^b = b*log a

==> log2 (x/4) = log2 x - 2log2 2

But log2 2 = 1

==> log2 (x/4) = log2 x - 2

Now we will rewrite log2 x = log4 x / log4 2

                                           = log4 x / log4 (4^1/2)

                                           = log4 x / 1/2 = 2log4 x

==> log2 (x/4) = 2log4 x -2

\But log4 x = 12

==> log2 (x/4) = 2*12 -2 = 22

==> log2 (x/4) = 22

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