Compare the graphs of the logarithmic functions f(x)= log(7)x and g(x) = log(4)x.

Question

For what values of x is f=g, f>g, and f<g? Explain how you know.

I already know what x is for f=g and f>g, but what is f<g? And how do I explain all this?

Answer 1

For easier comparison, let's have the same base for both log functions.
recall : log _a b = log b/loga or lnb/lna

so f(x) = log₇x = logx / log7
= (1/log7) logx
= appr 1.18329.. logx

g(x) = log₄ = logx/log4
= appr 1.66096.. logx

Assuming you got x=1 giving you equality of the two,

remember that log(any number > 1) is positive,
and log7 > log4, and logx would be the same for both.
What division result would give you a larger answer ?