Question

Given real numbers a,b > 1.

True of false? The exponential functions a^x and b^x are of the same order exactly when a≤b.

Answer 1

FALSE

Explanation:

Recall that a function f(x) is of exponential order c, if there exists a constant M such that and a real r such that

`|f(x)|\leq Me^(cx)\;(x\geq r)`

Now, take a = 2.5 and b = 2

The functions

`f(x)=(2.5)^x\;g(x)=2^x`

are both exponential of order 1, since

`\lim_(x \to \infty)(f(x))/(e^x)=\lim_(x \to \infty)(g(x))/(e^x)=0`

but a>b