ABC26GN2050 · Logarithms

$$\begin{aligned} & \log _{10} 2+16 \log _{10} \frac{16}{15}+12 \log _{10} \frac{25}{24}+7 \log _{10} \frac{81}{80} \\ & =\log _{10} 2+\log _{10}\left(\frac{16}{15}\right)^{16}+\log _{10}\left(\frac{25}{24}\right)^{12}+\log _{10}\left(\frac{81}{80}\right)^{7} \\ & =\log _{10}\left[2 \times\left(\frac{16}{15}\right)^{16} \times\left(\frac{25}{24}\right)^{12} \times\left(\frac{81}{80}\right)^{7}\right] \\ & =\log _{10}\left[2 \times \frac{\left(2^{4}\right)^{16}}{5^{16} \times 3^{16}} \times \frac{\left(5^{2}\right)^{12}}{\left(2^{3}\right)^{12} \times 3^{12}} \times \frac{\left(3^{4}\right)^{7}}{\left(2^{4}\right)^{7} \times 5^{7}}\right] \\ & =\log _{10}\left[\frac{2 \times 2^{64} \times 5^{24} \times 3^{28}}{5^{16} \times 3^{16} \times 2^{36} \times 3^{12} \times 2^{28} \times 5^{7}}\right] \end{aligned}$$