# Reduce to the simplest form: log2(24)/log96(2) -log2(192)/log12(2)

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We have to simplify log(2) 24 / log(96) 2 - log(2) 192 / log(12) 2

log(2) 24 / log(96) 2 - log(2) 192 / log(12) 2

use the property of logarithms: log(a) b = 1/log(b) a

=> log(2) 24* log(2) 96 - log(2) 192 * log(2) 12

=> log(2) 12*2 * log(2) 96 - log(2) 96*2 * log(2) 12

=> [log(2) 12 + log(2) 2]* log(2) 96 - [log(2) 96 + log(2) 2]* log(2) 12

=> log(2) 12 * log(2) 96 + log(2) 2 * log(2) 96 - log(2) 96 * log(2) 12 + log(2) 2* log(2) 12

=> log(2) 12 * log(2) 96 + log(2) 96 - log(2) 96 * log(2) 12 - log(2) 12

=> log(2) 96 - log(2) 12

=> log(2) [ 96/12]

=> log(2) 8

=> 3

**The simplified form of log(2) 24 / log(96) 2 - log(2) 192 / log(12) 2 = 3**

We'll write log 2 (24), using the product rule of logarithms:

log 2 (24) = log 2 (2) + log 2 (12)

We'll write log 96 (2) = 1/ log 2 (96)

log 2 (96) = 1 + log 2 (2) + log 2 (2) + log 2 (12)

log 2 (96) = 3 + log 2 (12)

log 2 (192) = 4 + log 2 (12)

log 12 (2) = 1/ log 2 (12)

We'll re-qrite the expression:

E = [1 + log 2 (12)][3 + log 2 (12)] - [4 + log 2 (12)]*log 2 (12)

We'll remove the brackets and we'll replace log 2 (12) by t:

E = 3 + 4t + t^2 - 4t - t^2

We'll combine and eliminate like terms:

E = 3

**The given expression reduced to the simplest form is E = 3.**