I am studying for an end of year exam and I completely forgot everything about logs and log functions (except for the extreme basics) so can someone tell me why the answer is D by showing me all the steps?

Any help would be greatly appreciated!

2. $10log a=log(a^{10})$
$-3log b= -log(b^3)$
$\frac{1}{2}log(9)=log(9^{\frac{1}{2}})=log \sqrt{9}=log(3)$

So...

$10log(a)-3log(b)+\frac{1}{2}log(9)=log(a^{10})-log(b^3)+log(3)$

And 2 of the log rules are:

$log(a)+log(b)=log(ab)$ and $log(c)-log(d)=log\left(\frac{c}{d}\right)$

So...

$10log(a)-3log(b)+\frac{1}{2}log(9)=log(a^{10})-log(b^3)+log(3)=log\left(\frac{a^{10}}{b^3}\right) +log(3)$

$=log\left(\frac{3a^{10}}{b^3}\right)$

3. log( x*y ) = log(x) + log(y)

log( x/y ) = log(x) - log(y)

log( x^n ) = n*log(x)

n*log(x) = log( x^n )

so...

log( 3*(a^10) / b^3 ) = log( 3*( a^10 ) ) - log( b^3 ) =

= log( 3 ) + log( a ^10 ) - 3 log( b ) =

= 2*( 1 / 2 )*log( 3 ) + 10log( a ) - 3log( b ) =

= (1/2)*log( 3^2 ) + 10log( a ) - 3log( b ) =

= (1/2)*log( 9 ) + 10log( a ) - 3log( b )