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Question about logs
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!
Thanks in advance!

$\displaystyle 10log a=log(a^{10})$
$\displaystyle 3log b= log(b^3)$
$\displaystyle \frac{1}{2}log(9)=log(9^{\frac{1}{2}})=log \sqrt{9}=log(3)$
So...
$\displaystyle 10log(a)3log(b)+\frac{1}{2}log(9)=log(a^{10})log(b^3)+log(3)$
And 2 of the log rules are:
$\displaystyle log(a)+log(b)=log(ab)$ and $\displaystyle log(c)log(d)=log\left(\frac{c}{d}\right)$
So...
$\displaystyle 10log(a)3log(b)+\frac{1}{2}log(9)=log(a^{10})log(b^3)+log(3)=log\left(\frac{a^{10}}{b^3}\right) +log(3)$
$\displaystyle =log\left(\frac{3a^{10}}{b^3}\right)$

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 )