Math Problem Solver

\[7 \cdot \log_{10}\left( \frac{10}{9} \right) - 2 \cdot \log_{10}\left( \frac{25}{24} \right) + 3 \cdot \log_{10}\left( \frac{81}{81} \right) = - \frac{7 \log{\left(9 \right)}}{\log{\left(10 \right)}} - \frac{2 \log{\left(\frac{25}{24} \right)}}{\log{\left(10 \right)}} + 7 \approx 0.28484490000386270459705410818355\]

\[\begin{aligned}7 \cdot \log_{10}\left( \frac{10}{9} \right) - 2 \cdot \log_{10}\left( \frac{25}{24} \right) + 3 \cdot \log_{10}\left( \frac{81}{81} \right)& = 7 \cdot \log{\left(\frac{10}{9} \right)} - 2 \cdot \log{\left(\frac{25}{24} \right)} + 3 \cdot \log{\left(\frac{81}{81} \right)}\\& = 7 \cdot \left(1 - \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(9 \right)}\right) - 2 \cdot \log{\left(\frac{25}{24} \right)} + 3 \cdot \log{\left(\frac{81}{81} \right)}\\& = \left(7 - 7 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(9 \right)}\right) - 2 \cdot \log{\left(\frac{25}{24} \right)} + 3 \cdot \log{\left(\frac{81}{81} \right)}\\& = \left(7 - 7 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(9 \right)}\right) - 2 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(\frac{25}{24} \right)} + 3 \cdot \log{\left(\frac{81}{81} \right)}\\& = \left(7 - 7 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(9 \right)}\right) - 2 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(\frac{25}{24} \right)} + 3 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(\frac{81}{81} \right)}\\& = \left(7 - 7 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(9 \right)} - 2 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(\frac{25}{24} \right)}\right) + 3 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(\frac{81}{81} \right)}\\& = 7 - 7 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(9 \right)} - 2 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(\frac{25}{24} \right)} + 3 \cdot \frac{1}{\log{\left(10 \right)}} \cdot \log{\left(\frac{81}{81} \right)}\end{aligned}\]