Foundations, part 1

$\begin{aligned} x & = 3\log \frac {81}{80} + 5 \log \frac {25}{24} + 7 \log \frac {16}{15} \\ & = \log \left( \frac {81}{80} \right)^3 + \log \left( \frac {25}{24} \right)^5 + \log \left( \frac {16}{15} \right)^7 \\ & = \log \left( \frac {81^3\cdot 25^5 \cdot 16^7}{80^3\cdot 24^5 \cdot 15^7} \right) \\ & = \log \left( \frac {3^{12}\cdot 5^{10} \cdot 2^{28}}{2^{12}5^3\cdot 2^{15}3^5 \cdot 3^75^7} \right) \\ & = \log \left( \frac {2^{28} \cdot 3^{12}\cdot 5^{10}}{2^{27}\cdot 3^{12}\cdot 5^{10}} \right) \\ & = \boxed{\log 2} \end{aligned}$

Given expression can be written as $\log \left(\dfrac{81}{80}\right)^3 + \log \left(\dfrac{25}{24}\right)^5 + \log \left(\dfrac{16}{15}\right)^7$

$\implies\log \left[\dfrac{3^{12} \times 5^{10} \times 2^{28}}{2^{12} \times 5^3 \times 2^{15} \times 3^5 \times 3^7 \times 5^7}\right]$

$\implies\log 2$