The rules for combining logs are

\[\begin{eqnarray*}&1.&  \log_b(ac)&=&\log_b(a)+\log_b(c)\\ \\ &2.&  \log_b\left(\frac{a}{c}\right)&=&\log_b(a)-\log_b(c)\\ \\ &3.&  \log_b(a^r)&=&r\log_b(a) \end{eqnarray*} \]

We see that:

\[\begin{eqnarray*}\log_{\var{b}}(\var{a_1})-\var{r_1}\log_{\var{b}}(\var{a_2})+\var{r_2}\log_{\var{b}}(\var{a_3})&=&\log_{\var{b}}(\var{a_1})-\log_{\var{b}}(\var{a_2}^{\var{r_1}})+\log_{\var{b}}(\var{a_3}^{\var{r_2}})\mbox{ using 3.}\\&=&\log_{\var{b}}(\var{a_1})-\log_{\var{b}}(\var{a_2^r_1})+\log_{\var{b}}(\var{a_3^r_2})\\&=&\log_{\var{b}}(\var{a_1}\times \var{a_3^r_2})-\log_{\var{b}}(\var{a_2^r_1}) \mbox{ using 1.}\\&=&\log_{\var{b}}\left(\frac{\var{a_1}\times \var{a_3^r_2}}{\var{a_2^r_1}}\right) \mbox{ using 2.}\\&=&\log_{\var{b}}\left(\frac{\var{a_1*a_3^r_2}}{\var{a_2^r_1}}\right)\\&=&\log_{\var{b}}\left(\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\right)\mbox{ on cancelling common factors}.\end{eqnarray*}\]

Hence $\displaystyle c=\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}$.

To calculate $\displaystyle \log_{\var{b}}\left(\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\right)$ to 4 decimal places we use the fact that for any positive base $b$:

\[\log_b(c)=\frac{\ln(c)}{\ln(b)}=\frac{\log_{10}(c)}{\log_{10}(b)}\]

and we can use either of the log functions, $\ln$ or $\log_{10}$ on our calculators to find the value.

Using $\ln$ we find:

\[ \log_{\var{b}}\left(\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\right)=\frac{\ln\left(\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\right)}{\ln(\var{b})}=\var{ans}\] to 4 decimal places.