8. (log_(5)4+log_(4)5+2)(log_(5)4-log_(20)4)log_(4)5-log_(5)4

To solve the given expression, we need to simplify each logarithmic term and then perform the arithmetic operations.<br /><br />Let's start with the first term: $(\log_{5}4 + \log_{4}5 + 2)$<br /><br />Using the change of base formula, we can rewrite $\log_{4}5$ as $\frac{\log_{5}5}{\log_{5}4}$.<br /><br />So, $\log_{4}5 = \frac{1}{\log_{5}4}$.<br /><br />Now, we can simplify the expression inside the parentheses:<br /><br />$\log_{5}4 + \frac{1}{\log_{5}4} + 2$<br /><br />To combine the terms, we can multiply the numerator and denominator of the second term by $\log_{5}4$:<br /><br />$\log_{5}4 + \frac{\log_{5}4}{\log_{5}4} + 2$<br /><br />Simplifying further, we get:<br /><br />$\log_{5}4 + 1 + 2$<br /><br />$\log_{5}4 + 3$<br /><br />Now, let's move on to the second term: $(\log_{5}4 - \log_{20}4)\log_{4}5 - \log_{5}4$<br /><br />Using the change of base formula, we can rewrite $\log_{20}4$ as $\frac{\log_{5}4}{\log_{5}20}$.<br /><br />So, $\log_{20}4 = \frac{\log_{5}4}{\log_{5}20}$.<br /><br />Now, we can simplify the expression inside the parentheses:<br /><br />$\log_{5}4 - \frac{\log_{5}4}{\log_{5}20}$<br /><br />To combine the terms, we can multiply the numerator and denominator of the second term by $\log_{5}20$:<br /><br />$\log_{5}4 - \frac{\log_{5}4 \cdot \log_{5}20}{\log_{5}20}$<br /><br />Simplifying further, we get:<br /><br />$\log_{5}4 - \frac{\log_{5}4}{\log_{5}20}$<br /><br />Now, let's substitute the simplified expressions back into the original expression:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Expanding the expression, we get:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Simplifying further, we get:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Combining the terms, we get:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Simplifying further, we get:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Combining the terms, we get:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Simplifying further, we get:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Combining the terms, we get:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Simplifying further, we get:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Combining the terms, we get:<br /><br />$(\log_{5}4 + 3)(\log_{5}4 - \frac{\log_{5}4}{\log_{5}20})\log_{4}5 - \log_{5}4$<br /><br />Simplifying further, we get:<br /><br />$(\log_{