We are given:
\(\displaystyle{5}{\left({10}^{{x}}−{6}\right)}={7}\)

Divide both sides by 5: \(\displaystyle{10}^{{x}}-{6}={1.4}\)

Take the (common) logarithm of both sides: \(\displaystyle{{\log{{10}}}^{{x}}-}{6}={\log{{1.4}}}\)

Recall that \(\displaystyle{\log{{b}}}\cdot{b}^{{x}}={x}\) and since the common logarithm is base 10, we write: \(\displaystyle{x}-{6}={\log{{1.4}}}\)

Add 6 to both sides: \(\displaystyle{x}={6}+{\log{{1.4}}}\sim{6.146}\)

Divide both sides by 5: \(\displaystyle{10}^{{x}}-{6}={1.4}\)

Take the (common) logarithm of both sides: \(\displaystyle{{\log{{10}}}^{{x}}-}{6}={\log{{1.4}}}\)

Recall that \(\displaystyle{\log{{b}}}\cdot{b}^{{x}}={x}\) and since the common logarithm is base 10, we write: \(\displaystyle{x}-{6}={\log{{1.4}}}\)

Add 6 to both sides: \(\displaystyle{x}={6}+{\log{{1.4}}}\sim{6.146}\)