Question

Which expression is equivalent to log12 x^4sqrt x^3-2/(x+1)^5
A 4log12 x+1/2log12 (x^3-2)-5log12 (x×1)

B 4log12x+1/2log12x^3/2-5log12x+log12 1

C Log124x+1/2log12 (x^3-2)-5log12 (x)+1

D 4Log12x+1/2log12 (x^3-2)-5log12 (x+1)

Answer 1

The given expression is:

$log_{12}( \frac{x^{4} \sqrt{ x^{3}-2 } }{(x+1)^{5}} )$

Using the rules of log:

$log(ab)=log(a)+log(b) \\ \\ log( \frac{a}{b} )=log(a)-log(b) \\ \\ log(a)^{x}=xlog(a)$

We can simplify the given expression as:

$log_{12}( x^{4} ) +log_{12}( \sqrt{ x^{3}-2 } ) -log_{12}((x+1)^{5} ) \\ \\ =log_{12}( x^{4} ) +log_{12}((x^{3}-2)^{ \frac{1}{2} } ) -log_{12}((x+1)^{5} ) \\ \\ =4log_{12}(x)+ \frac{1}{2} log_{12}( x^{3}-2 )-5log_{12}(x+1)$

This is the simplified form the of the given expression. 

Thus, option D gives the correct answer