How does the base of a logarithm affect a graph?

Question

I know what changing the base of a logarithm does graphically, but I'm having trouble trying to explain it mathematically

Answer 1

y = f(x)

what y is depends on what x is.

when f(x) = log x (common log, i.e. base 10), obviously most of the values differ from g(x) = ln x (base e).

But the x intercepts are the same, because ln 1 = log 1 = 0 regardless of base.

In all other cases, the f(x) and g(x) values differ, and that should not be shocking.