Swapping Positions

Relevant wiki: Logarithms

Using the identity $\log_{a} {b} =\dfrac{1}{\log_{b}{a}}$ we see:

$\log_{7}{10}=\dfrac{1}{\log_{10}{7}} \implies \log_{7}{10} \times \log_{10}{7}=1$

$\boxed{\boxed{1}}$

$\log_{10}7 \times \log_7 10\\ =\dfrac{\log_7 7}{\cancel{\log_7 10}} \times \cancel{\log_7 10}\\ =\log_7 7\\ =\boxed{1}$

Change of base:

$\log_b a = \frac {log a}{log b}$

$\log_{10} {7} = \frac {log {7}}{log {10}}$

$\log_{7} {10} = \frac {log {10}}{log {7}}$

$\frac {log {7}}{log {10}} \times \frac {log {10}}{log {7}} = 1$

$\begin{aligned} \log_{10} 7 \log_7 10 & = \log_7 10^{\log_{10} 7} \\ & = \log_7 7 \\ & = \boxed{1} \end{aligned}$ .