Final answer:
Both f(g(x)) and g(f(x)) simplify to x as log base 10 and 10 to the power of are inverse functions, resulting in the original input value x for the composition of the two.
Step-by-step explanation:
The question asks us to find f(g(x)) and g(f(x)) where f(x) is the common logarithm function (log base 10) and g(x) is the exponential function with base 10. To find f(g(x)), we substitute g(x) into f(x), which gives us f(10x) or log10(10x). By the property of logarithms, this simplifies to x, because the logarithm and exponential with the same base are inverse functions.
To evaluate g(f(x)), we substitute f(x) into g(x), resulting in 10log10(x). Again, since the functions are inverse, this simplifies to x. Therefore, both f(g(x)) and g(f(x)) are equal to x for all positive values of x.