Question

write as the sum and/or difference of logarithms. Express powers as factors.

Answer 1

\[\log_{6} \left( x-4/x^7 \right)\]

Answer 2

first combine the inside into one fraction with x^7 as common denominator
x - 4/x^7 = (x^8 - 4)/x^7
Now that we have division we split it into 2 logs using the log rule:
logx - logy = log(x/y) *note the base does not change
=>log(x^8 - 4) - log(x^7)
Another rule is log(x^n) = n*log(x) so change log(x^7) to 7log(x)
=> log(x^8 -4) - 7log(x)
Now we can factor x^8 -4 into a squared binomial: (x^4+2)(x^4-2)
=>log((x^4+2)*(x^4-2)) - 7log(x)
Now we have multiplication in our log so we split that into 2 logs
logx + logy = logxy
=>log(x^4+2) + log(x^4-2) - 7log(x)
ok now i hope you get the idea, these could be factored again and split up and so on