How do you simplify #\log _ { 2} ( \frac { x ^ { 6} y ^ { 7} } { 8} )#?

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Answer 1 · 2017-06-15T02:47:20

The expression equals #6log^2x + 7log^2y - 3#

Use the following laws of logarithms to simplify the above problem

#• log_a(nm) = log_a n + log_a m#
#•log_a(n/m) = log_an - log_a m#
#•log(a^n) = nloga#

Now apply them:

#=log_2(x^6y^7) - log_2 8#

#= log_2 x^6 + log_2 y^7 - 3#

#=6log_2x + 7log_2y - 3#

Hopefully this helps!

Answer 2 · 2017-06-15T02:54:39

#log_2(x^6)+log_2(y^7)-3#

Use log properties:
Product Rule:
#logx+logy=logxy#

Quotient Rule:
#logx-logy=log(x/y)#

Split the fraction into two separate logs using the Quotient Rule.
#log_2((x^6y^7)/8)#
#log_2(x^6y^7)-log_2(8)#

Split #(x^6y^7)# into two separate logs using the Product Rule.
#log_2(x^6y^7)-log_2(8)#
#log_2(x^6)+log_2(y^7)-log_2(8)#

Simplify #-log_2(8)#.
#log_2(x^6)+log_2(y^7)-log_2(8)#
#log_2(x^6)+log_2(y^7)-3#

#log_2(x^6)+log_2(y^7)-3# is your simplified expression.