Calculate log (15) - 2 log (5/4) + log (100/243)

$$2 \log \left(\frac{5}{4}\right) = \log \left(\left(\frac{5}{4}\right)^2\right) = \log \left(\frac{25}{16}\right)$$

The expression becomes:

$$\log (15) - \log \left(\frac{25}{16}\right) + \log \left(\frac{100}{243}\right)$$

Combine the terms into a single logarithm:

$$\log \left( \frac{15}{\frac{25}{16}} \times \frac{100}{243} \right)$$ $$\log \left( 15 \times \frac{16}{25} \times \frac{100}{243} \right)$$

$$\log \left( \frac{15 \times 16 \times 100}{25 \times 243} \right)$$

Cancel common factors:

$$\log \left( \frac{(3 \times 5) \times 16 \times (4 \times 25)}{25 \times (3 \times 81)} \right)$$

Cancel 5 and 25 (from $15/25$):

$$\log \left( \frac{3 \times 16 \times 100}{5 \times 243} \right)$$

Cancel 5 and 100:

$$\log \left( \frac{3 \times 16 \times 20}{243} \right)$$

Cancel 3 and 243:

$$\log \left( \frac{16 \times 20}{81} \right)$$ $$\log \left( \frac{320}{81} \right)$$

The final answer is $\boxed{\log \left(\frac{320}{81}\right)}$.

: Scientific Notation Calculation
Calculate $\frac{(0.00042 \times 10^{-8})(15,000)}{(5000 \times 10^7)(0.0021 \times 10^{14})}$

$$0.00042 = 4.2 \times 10^{-4}$$ $$15,000 = 1.5 \times 10^4$$ $$5000 = 5 \times 10^3$$ $$0.0021 = 2.1 \times 10^{-3}$$

Substitute these into the expression:

$$\frac{(4.2 \times 10^{-4} \times 10^{-8})(1.5 \times 10^4)}{(5 \times 10^3 \times 10^7)(2.1 \times 10^{-3} \times 10^{14})}$$

Numerator:

$$(4.2 \times 1.5) \times (10^{-4} \times 10^{-8} \times 10^4) = 6.3 \times 10^{(-4 - 8 + 4)} = 6.3 \times 10^{-8}$$

Denominator:

$$(5 \times 2.1) \times (10^3 \times 10^7 \times 10^{-3} \times 10^{14}) = 10.5 \times 10^{(3 + 7 - 3 + 14)} = 10.5 \times 10^{21}$$

The expression becomes:

$$\frac{6.3 \times 10^{-8}}{10.5 \times 10^{21}}$$

Divide the numerical parts:

$$\frac{6.3}{10.5} = \frac{63}{105}$$

Both 63 and 105 are divisible by 21: $63 \div 21 = 3$ and $105 \div 21 = 5$.

$$\frac{3}{5} = 0.6$$

Divide the powers of 10:

$$\frac{10^{-8}}{10^{21}} = 10^{-8 - 21} = 10^{-29}$$

Combine these results:

$$0.6 \times 10^{-29}$$

$$0.6 \times 10^{-29} = (6 \times 10^{-1}) \times 10^{-29} = 6 \times 10^{(-1 - 29)} = 6 \times 10^{-30}$$

The final answer is $\boxed{6 \times 10^{-30}}$.