How do you simplify \log _ { 2} ( \frac { x ^ { 6} y ^ { 7} } { 8} )log2(x6y78)?

2 Answers
Jun 15, 2017

The expression equals 6log^2x + 7log^2y - 36log2x+7log2y3

Explanation:

Use the following laws of logarithms to simplify the above problem

• log_a(nm) = log_a n + log_a mloga(nm)=logan+logam
•log_a(n/m) = log_an - log_a mloga(nm)=loganlogam
•log(a^n) = nlogalog(an)=nloga

Now apply them:

=log_2(x^6y^7) - log_2 8=log2(x6y7)log28

= log_2 x^6 + log_2 y^7 - 3=log2x6+log2y73

=6log_2x + 7log_2y - 3=6log2x+7log2y3

Hopefully this helps!

Jun 15, 2017

log_2(x^6)+log_2(y^7)-3log2(x6)+log2(y7)3

Explanation:

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

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

Split the fraction into two separate logs using the Quotient Rule.
log_2((x^6y^7)/8)log2(x6y78)
log_2(x^6y^7)-log_2(8)log2(x6y7)log2(8)

Split (x^6y^7)(x6y7) into two separate logs using the Product Rule.
log_2(x^6y^7)-log_2(8)log2(x6y7)log2(8)
log_2(x^6)+log_2(y^7)-log_2(8)log2(x6)+log2(y7)log2(8)

Simplify -log_2(8)log2(8).
log_2(x^6)+log_2(y^7)-log_2(8)log2(x6)+log2(y7)log2(8)
log_2(x^6)+log_2(y^7)-3log2(x6)+log2(y7)3

log_2(x^6)+log_2(y^7)-3log2(x6)+log2(y7)3 is your simplified expression.