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

2 Answers
Jun 15, 2017

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

Explanation:

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!

Jun 15, 2017

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

Explanation:

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.