logarithms

log_a s  = 5    log_a t   =10

In exponential form, we have

a^5  = s       and  a^10  =  t

So

log_a (s / t^2 )  =

log_a [a^5 / (a^10)^2 ]  =

log_a [ a^5 / a^20 ]  =

log_a  a^(-15)        and by a log property we can write

-15 * log_a a  =

 ( Note  log_a a   =  1  )

-15 * 1  =

-15

If loga(s) = 5 and loga(t) = 10, then loga(s/t2) = ?

$\begin{array}{|rcll|} \hline \log_a(s) &=& 5 \\ \log_a(t) &=& 10 \\\\ \log_a(s) - 2\cdot \log_a(t) &=& 5 - 2\cdot 10 \\ \log_a(s) - 2\cdot \log_a(t) &=& -15 \\ \log_a(s) -\log_a(t^2) &=& -15 \\\\ \mathbf{\log_a\left(\dfrac{s}{t^2} \right)} & \mathbf{=} & \mathbf{-15} \\ \hline \end{array}$