Evaluate:

$\log \space 5 + 16 \log \space \Big(\dfrac{625}{6}\Big) + 12 \log \space \Big(\dfrac{4}{375}\Big) + 7 \log \space \Big(\dfrac{81}{1250}\Big)$

Given,

$\Rightarrow \log \space 5 + 16 \log \space \Big(\dfrac{625}{6}\Big) + 12 \log \space \Big(\dfrac{4}{375}\Big) + 7 \log \space \Big(\dfrac{81}{1250}\Big) \\[1em] \Rightarrow \log \space 5 + \log \space \Big(\dfrac{625}{6}\Big)^{16} + \log \space \Big(\dfrac{4}{375}\Big)^{12} + \log \space \Big(\dfrac{81}{1250}\Big)^{7} \\[1em] \Rightarrow \log \space 5 + \log \space \Big(\dfrac{5^4}{2 \times 3}\Big)^{16} + \log \space \Big(\dfrac{2^2}{3 \times 5^3 }\Big)^{12} + \log \space \Big(\dfrac{3^4}{2 \times 5^4}\Big)^{7} \\[1em] \Rightarrow \log \space 5 + \log \space \Big(\dfrac{5^{(4 \times 16)}}{2^{16} \times 3^{16}}\Big) + \log \space \Big(\dfrac{2^{2 \times 12}}{3^{12} \times 5^{3 \times 12}}\Big) + \log \space \Big(\dfrac{3^{4 \times 7}}{2^7 \times 5^{4 \times 7}}\Big) \\[1em] \Rightarrow \log \space 5 + \log \space \Big(\dfrac{5^{64}}{2^{16} \times 3^{16}}\Big) + \log \space \Big(\dfrac{2^{24}}{3^{12} \times 5^{36} }\Big) + \log \space \Big(\dfrac{3^{28}}{2^{7} \times 5^{28}}\Big) \\[1em] \Rightarrow \log \space \Big[\dfrac{5 \times 5^{64} \times 2^{24} \times 3^{28} }{2^{16} \times 3^{16} \times 3^{12} \times 5^{36} \times 2^{7} \times 5^{28}}\Big] \\[1em] \Rightarrow \log \space \Big[\dfrac{5^{64 + 1} \times 2^{24} \times 3^{28} }{2^{16 + 7} \times 3^{16 + 12} \times 5^{36 + 28}}\Big] \\[1em] \Rightarrow \log \space \Big[\dfrac{5^{65} \times 2^{24} \times 3^{28} }{2^{23} \times 3^{28} \times 5^{64}}\Big] \\[1em] \Rightarrow \log \space [5^{65 - 64} \times 2^{24 - 23} \times 3^{28 - 28} ] \\[1em] \Rightarrow \log \space [5^1 \times 2^1 \times 3^0] \\[1em] \Rightarrow \log \space [5 \times 2 \times 1] \\[1em] \Rightarrow \log \space 10 \\[1em] \Rightarrow 1.$

Hence, $\log \space \Big(\dfrac{81}{8}\Big) + 2 \log \space \dfrac{2}{3} − 3 \log \space \dfrac{3}{2} + \log \space \dfrac{3}{4}$ = 1.