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

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have log(4) x = 12. We need to find log(2) (x/4)

log(2) (x/4) = log (2) x - log(2) 4

=> log(2) x - log(2) 2^2

use log a^b = a*log b

=> log(2) x - 2 ...(1)

log(4) x = 12

=> x = 4^12

=> x = 2^2^12

=> x = 2^24

take log to base 2 of both the sides

=> log(2) x = 24

Substituting in (1)

log(2) x - 2

=> 24 - 2

=> 22

The required value of log(2)(x/4) = 22

hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

log4 (x) = 12

We will use logarithm properties to simplify.

We will rewrite:

 ==> log4 x = log2 x/log2 4 = log2 x / 2 = 12

 ==> log2 x = 2*12 = 24

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

                     = log2 x - log2 2^2

                       = log2 x - 2

                         = 24 -2 = 22

==> log2 (x/4) = 22

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll consider the constraint and we'll take antilogarithm:

log4 (x) = 12 => x = 4^12

We'll substitute x by 4^12 in the fraction:

log2 (x/4) = log2 (4^12/4)

Since the exponentials have matching bases, we'll subtract exponents:

4^12/4 = 4^(12-1) = 4^11

log2 (4^12/4) = log2 (4^11)

We'll use power rule of logarithms:

log2 (4^11) = 11log2 (4)

But 4 = 2^2

11log2 (2^2) = 22 log2 (2)

But log2 (2) = 1 => 22 log2 (2) = 22

Therefore, log2 (x/4) = 22.

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