Who can give a strict proof of monotonicity of exponential function? Can you be more detailed I just want to avoid using high numbers Let X1 be less than X2, that is, △ x = x2-x1 > 0 ∴△y＝a^X2-a^X1=a^X1*[a^(X2-X1)-1] When a ﹥ 1, as long as we can prove that a ^ (x2-x1) ﹥ 1, we can prove that the function is A > 1 is an increasing function. So the problem now becomes how to prove that when a > 1, there is a ^ (x2-x1) > 1. This seems to involve the monotonicity of power function. How to deal with power function? If only anyone knew (try to avoid Advanced Mathematics)

Who can give a strict proof of monotonicity of exponential function? Can you be more detailed I just want to avoid using high numbers Let X1 be less than X2, that is, △ x = x2-x1 > 0 ∴△y＝a^X2-a^X1=a^X1*[a^(X2-X1)-1] When a ﹥ 1, as long as we can prove that a ^ (x2-x1) ﹥ 1, we can prove that the function is A > 1 is an increasing function. So the problem now becomes how to prove that when a > 1, there is a ^ (x2-x1) > 1. This seems to involve the monotonicity of power function. How to deal with power function? If only anyone knew (try to avoid Advanced Mathematics)

For a ^ x, a > 0, we can't discuss its monotonicity without explaining its exact definition
Exponential function is defined on the whole real number interval
A ^ n = a * a *... * a (n > 0, the same below) (multiply n a)
a^0 = 1
a^(-n) = 1 / a^n
Let's talk about the definition on rational number set
A ^ (1 / N) = the nth arithmetic root of a,
A ^ (P / Q) = (a ^ P), where P / Q is the reduced fraction
In fact, for a ^ (P1 / Q1) and a ^ (P2 / Q2), we can divide the fractions P1 / Q1 and P2 / Q2, so that the denominator is the same, let them be P1 '/ Q and P2' / Q respectively. Now we are comparing the two numbers with a ^ (1 / Q) as the base and P1 'and P2' as the index When a < 1, a ^ (1 / Q) > 1, then we can know that the function is strictly monotone increasing; conversely, when a < 1, we can also prove that the function is strictly monotone decreasing
Now for any real number x, we can take a rational sequence {QN}. When n increases infinitely, it tends to x monotonically. Then we can define a ^ x as the limit of a ^ QN when n increases infinitely
If we use rational number sequence to approximate exponential function, then we can take two rational numbers with enough accuracy to replace two real numbers whose difference is arbitrarily close to each other, and keep the size relationship unchanged. (of course, there are some operational guarantees, such as why the definition is reasonable, that is, why the limit exists, and so on, I will not write.), The monotonicity of exponential function under real number can be reduced to the monotonicity under rational number

Proof of monotonicity of exponential function by quotient method
The x power of y = a

Let x 1, x 2 belong to R, and x 1

Strict proof of monotonicity of exponential function

Proof: Let f (x) = x times of a, a > 0, X ∈ R
F '(x) = x power of a * LNA
① If a > 1, then LNA > 0, then f '(x) > 0, the exponential function increases monotonically
② If a ＜ 1, then LNA ＜ 0, then f '(x) ＜ 0, the exponential function point drops and decreases
It's over