Answer
$$1$$
Work Step by Step
$$\eqalign{
& \int_1^e {\frac{{2\ln 10{{\log }_{10}}x}}{x}} dx \cr
& {\text{use constant multiple rule}} \cr
& = 2\ln 10\int_1^e {\frac{{{{\log }_{10}}x}}{x}} dx \cr
& {\text{using the property }}{\log _a}x = \frac{{\ln x}}{{\ln a}}{\text{ }}\left( {{\text{see example 7b}}} \right) \cr
& = 2\ln 10\int_1^e {\frac{{\ln x}}{{x\ln 10}}} dx \cr
& = 2\int_1^e {\frac{{\ln x}}{x}} dx \cr
& {\text{integrate using the power rule }}\int {{u^n}du = \frac{{{u^{n + 1}}}}{{n + 1}} + C} \cr
& {\text{for this exercise, we can note that }}u = \ln x{\text{; then:}} \cr
& = 2\left( {\frac{{{{\left( {\ln x} \right)}^2}}}{2}} \right)_1^e \cr
& = \left( {{{\left( {\ln x} \right)}^2}} \right)_1^e \cr
& {\text{use fundamental theorem of calculus: }}\cr
& \int_a^b {f\left( x \right)} dx = F\left( b \right) - F\left( a \right).\,\,\left( {{\text{see page 281}}} \right) \cr
& = {\left( {\ln e} \right)^2} - {\left( {\ln 1} \right)^2} \cr
& {\text{simplifying}} \cr
& = {1^2} - {0^2} \cr
& = 1 \cr} $$
