Question

combine the expression 1/3 log (2x+1) +1/2[log(x-4)-log(x^4-x^2-1]

Answer 1

well you need 2 log laws to simplify

law 1 powers

\[alog(b) = \log(b^a)\]

law 2 multiplication

\[\log(a) + \log(b) = \log(ab)\]

law 3 division

\[\log(a) - \log(b) = \log(\frac{a}{b})\]

looking at the right hand brackets 1st

\[\frac{1}{3}\log(2x + 1)+ \frac{1}{2}(\log(\frac{(x -4)}{\log(x^4 - x^2 -1)})\]

now do law 1

\[\log(2x + 1)^{\frac{1}{3}} + \log(\frac{x -4}{x^4 -x^2 -1})^{\frac{1}{2}}\]

you can simplify this line by changing the fractions indices to radicals.

\[\log (\sqrt[3]{2x + 1)} + \log(\sqrt{\frac{x -4}{x^4 -x^2 -1}})\]

I'll leave you to do the multiplication law

Answer 2

@campbell_st so do we just put everying togehter or multiply together?

Answer 3

well you just combine then as 1 log

the answer really depends on what form your answer needs to be

but 1 choice may be

\[lon(\sqrt[3]{2x +1}\sqrt{\frac{x -4}{x^3 -x^2 -1}})\]