Answer
$\dfrac{\log{7}}{\log{2}}$
Work Step by Step
RECALL:
The change-of-base formula for loagarithms:
$\log_b{x} = \dfrac{\log_a{x}}{\log_a{b}}$
Use the formula above to obtain:
$(\log_2{5})(\log_5{7}) = \left(\dfrac{\log{5}}{\log{2}}\right)\left(\dfrac{\log{7}}{\log{5}}\right)$
Cancel common factors to obtain:
$\require{cancel}=\left(\dfrac{\cancel{\log{5}}}{\log{2}}\right)\left(\dfrac{\log{7}}{\cancel{\log{5}}}\right)
\\=\dfrac{\log{7}}{\log{2}}$