Index and logarithm A ^ B = n, the number B is called the logarithm of n with a as the base Isn't the number B an exponent? Why is it called logarithm?

Index and logarithm A ^ B = n, the number B is called the logarithm of n with a as the base Isn't the number B an exponent? Why is it called logarithm?

B is the exponent of a and the logarithm of n with a as the base. Note that the two objects are different
The exponent is for a and the logarithm is for n

Let e be higher

Let f (x) = (LNX) ^ 2
f'(x) = 2lnx / x
According to the mean value theorem, there is a point C in the interval (a, b)
2lnc/c = [(Inb)^2-(Ina)^2]/(b-a)
F '(x) = 2lnx / X is a decreasing function in the interval (a, b)
f'(c) > f'(b) > f'(e^2) = 4/e^2
[(Inb)^2-(Ina)^2]/(b-a) > 4/e^2
(Inb)^2-(Ina)^2 > 4 (b-a)/(e^2)

Why is ina × inb equal to the inb power of ina

Certification:
Let B ^ (LNA) = X
So LNA = log (b) X
According to the bottom changing formula:
lna = (lnX)/lnb
lna lnb = lnX = ln b^(lna)