log_(5)(1)/(2)cdot log_(2)(1)/(7)cdot log_(7)(1)/(5)=

To solve the given expression, we can use the properties of logarithms.<br /><br />The expression is:<br />$log_{5}\frac {1}{2}\cdot log_{2}\frac {1}{7}\cdot log_{7}\frac {1}{5}$<br /><br />We can rewrite the expression as:<br />$log_{5}\frac {1}{2} + log_{2}\frac {1}{7} + log_{7}\frac {1}{5}$<br /><br />Now, we can use the change of base formula to rewrite the logarithms with a common base:<br />$log_{5}\frac {1}{2} = \frac{log_{10}\frac {1}{2}}{log_{10}5}$<br />$log_{2}\frac {1}{7} = \frac{log_{10}\frac {1}{7}}{log_{10}2}$<br />$log_{7}\frac {1}{5} = \frac{log_{10}\frac {1}{5}}{log_{10}7}$<br /><br />Substituting these values back into the expression, we get:<br />$\frac{log_{10}\frac {1}{2}}{log_{10}5} + \frac{log_{10}\frac {1}{7}}{log_{10}2} + \frac{log_{10}\frac {1}{5}}{log_{10}7}$<br /><br />Now, we can simplify the expression by combining the logarithms:<br />$\frac{log_{10}\frac {1}{2} + log_{10}\frac {1}{7} + log_{10}\frac {1}{5}}{log_{10}5 \cdot log_{10}2 \cdot log_{10}7}$<br /><br />Using the properties of logarithms, we can rewrite the numerator as:<br />$log_{10}\left(\frac{1}{2} \cdot \frac{1}{7} \cdot \frac{1}{5}\right)$<br /><br />Simplifying the numerator, we get:<br />$log_{10}\left(\frac{1}{70}\right)$<br /><br />Now, we can rewrite the denominator as:<br />$log_{10}5 \cdot log_{10}2 \cdot log_{10}7$<br /><br />Substituting the numerator and denominator back into the expression, we get:<br />$\frac{log_{10}\left(\frac{1}{70}\right)}{log_{10}5 \cdot log_{10}2 \cdot log_{10}7}$<br /><br />Using the properties of logarithms, we can rewrite the numerator as:<br />$-log_{10}(70)$<br /><br />Now, we can rewrite the denominator as:<br />$log_{10}5 \cdot log_{10}2 \cdot log_{10}7$<br /><br />Substituting the numerator and denominator back into the expression, we get:<br />$\frac{-log_{10}(70)}{log_{10}5 \cdot log_{10}2 \cdot log_{10}7}$<br /><br />Now, we can simplify the expression by canceling out the common factors in the numerator and denominator:<br />$\frac{-1}{log_{10}5 \cdot log_{10}2 \cdot log_{10}7}$<br /><br />Therefore, the final answer is:<br />$\boxed{\frac{-1}{log_{10}5 \cdot log_{10}2 \cdot log_{10}7}}$