Alternative Equation Formats

$\ln \left(\frac{p_{1\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}}{p_{3\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}}\right),\ln \left(\frac{p_{2\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}}{p_{3\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}}\right),\ln \left(\frac{p_{\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>1}}{p_{\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>3}}\right),\ln \left(\frac{p_{\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>2}}{p_{\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>3}}\right)$

$\ln \left(\frac{p_{1\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}}{p_{3\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}}\right),\ln \left(\frac{p_{2\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}}{p_{3\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>}}\right),\ln \left(\frac{p_{\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>1}}{p_{\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>3}}\right),\ln \left(\frac{p_{\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>2}}{p_{\cdot <!--\; \cdot \; --><span\; class="Comment\_comment\_\_N0+Yl"></span>3}}\right)$

$ \ln \left( \frac{p_{1 \cdot }}{p_{3 \cdot }} \right), ~ \ln \left( \frac{p_{2 \cdot }}{p_{3 \cdot }} \right), ~ \ln \left( \frac{p_{\cdot 1}}{p_{\cdot 3}} \right), ~ \ln \left( \frac{p_{\cdot 2}}{p_{\cdot 3}} \right) $