Simplify the logarithmic expression: 10((30)/(16)) - 210((5)/(9)) + 10((48)/(243))

$$2\log_{10}\left(\frac{5}{9}\right) = \log_{10}\left(\left(\frac{5}{9}\right)^2\right) = \log_{10}\left(\frac{25}{81}\right)$$

The original expression now becomes:

$$\log_{10}\left(\frac{30}{16}\right) - \log_{10}\left(\frac{25}{81}\right) + \log_{10}\left(\frac{48}{243}\right)$$

$$\log_{10}\left(\frac{\frac{30}{16}}{\frac{25}{81}} \times \frac{48}{243}\right)$$

This simplifies to:

$$\log_{10}\left(\frac{30}{16} \times \frac{81}{25} \times \frac{48}{243}\right)$$

First, simplify each individual fraction:
$\frac{30}{16} = \frac{15}{8}$
$\frac{48}{243} = \frac{16 \times 3}{81 \times 3} = \frac{16}{81}$
Substitute these simplified fractions back into the expression:

$$\log_{10}\left(\frac{15}{8} \times \frac{81}{25} \times \frac{16}{81}\right)$$

Cancel out common factors (81 in the numerator and denominator, and 8 with 16):

$$\log_{10}\left(\frac{15}{1} \times \frac{1}{25} \times \frac{2}{1}\right)$$

Multiply the remaining terms:

$$\log_{10}\left(\frac{15 \times 2}{25}\right)$$

Simplify further by dividing 15 and 25 by 5:

$$\log_{10}\left(\frac{3 \times 2}{5}\right)$$ $$\log_{10}\left(\frac{6}{5}\right)$$