7log10(10/9)-2log10(25/24)+3log10(81/81)

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on Nov 15, 2024 at 11:50 am



You asked:

Evaluate the expression: 7log⁡10(109)−2log⁡10(2524)+3log⁡10(8181)7 \log_{10}\left( \frac{10}{9} \right) - 2 \log_{10}\left( \frac{25}{24} \right) + 3 \log_{10}\left( \frac{81}{81} \right)

MathBot Answer:

7log⁡10(109)−2log⁡10(2524)+3log⁡10(8181)=−7log⁡(9)log⁡(10)−2log⁡(2524)log⁡(10)+7≈0.284844900003862704597054108183557 \log_{10}\left( \frac{10}{9} \right) - 2 \log_{10}\left( \frac{25}{24} \right) + 3 \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

7log⁡10(109)−2log⁡10(2524)+3log⁡10(8181)=7⋅log⁡(109)−2⋅log⁡(2524)+3⋅log⁡(8181)=7⋅(1−1log⁡(10)⋅log⁡(9))−2⋅log⁡(2524)+3⋅log⁡(8181)=(7−7⋅1log⁡(10)⋅log⁡(9))−2⋅log⁡(2524)+3⋅log⁡(8181)=(7−7⋅1log⁡(10)⋅log⁡(9))−2⋅1log⁡(10)⋅log⁡(2524)+3⋅log⁡(8181)=(7−7⋅1log⁡(10)⋅log⁡(9))−2⋅1log⁡(10)⋅log⁡(2524)+3⋅1log⁡(10)⋅log⁡(8181)=(7−7⋅1log⁡(10)⋅log⁡(9)−2⋅1log⁡(10)⋅log⁡(2524))+3⋅1log⁡(10)⋅log⁡(8181)=7−7⋅1log⁡(10)⋅log⁡(9)−2⋅1log⁡(10)⋅log⁡(2524)+3⋅1log⁡(10)⋅log⁡(8181)\begin{aligned}7 \log_{10}\left( \frac{10}{9} \right) - 2 \log_{10}\left( \frac{25}{24} \right) + 3 \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}

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