Given that the domain of the function f (x) = LG (ax kbx) (k > 0, a > 1 > b > 0) is exactly (0, + ∞), is there such a, B that f (x) takes a positive value on (1, + ∞), and F (3) = LG4? If it exists, find the values of a and B; If not, please explain the reason

Given that the domain of the function f (x) = LG (ax kbx) (k > 0, a > 1 > b > 0) is exactly (0, + ∞), is there such a, B that f (x) takes a positive value on (1, + ∞), and F (3) = LG4? If it exists, find the values of a and B; If not, please explain the reason

Solution ∵ ax kbx ＞ 0, i.e   (AB) x > K   A ＞ 1 ＞ B ＞ 0,  ab ＞ 1  x ＞ logabk is the condition satisfied by its definition field, and  function F   （x）   The definition field of is exactly (0, + ∞), ‡ logabk = 0, ‡ k = 1. ‡ F   (x) = LG (AX BX). If appropriate

Limtanx-sinx / SIN3 power x.x tends to infinity

When x → 0,
（tanx-sinx)/(sinx)^3
=(1-cosx)/[cosx(sinx)^2]
=(1-cosx0/{cosx[1-(cosx)^2]}
=1/[cosx(1+cosx)]
→1/2；
When x →∞, the limit does not exist

X power of 10 = 2, y power of 10 = 3, find the value of 2x quarter y of 100

∵10^x=2,10^y=3
∴100^(2x-y/4)=(100^2x)/[100^(y/4)]
=[(10^x)^4]/[(10^y)^(1/2)]
=(2^4)/[3^(1/2)]
=(16√3)/3

The third power of (X-Y) * (Y-X) the seventh power * (X-Y) the fourth power Multiplication and combination of powers of the same base

（x-y）^3*（y-x）^7*（x-y）^4
=(x-y)^(3+4)*(y-x)^7
=(x-y)^7*(y-x)^7
=[(x-y)(y-x)]^7
=[-(x-y)^2]^7
=-(x-y)^14

The number of digits of the 100th power of 3.76 * 10 is

3.76*10^100=3.76*10^2*10^98=376*10^98
Namely
3 + 98 = 101 bits

(X-Y) seventh power ÷ [(X-Y) third power ÷ (Y-X)] third power Another question is if (a) is to the power of M + 1 × N + 2 power of B) (2n-1 power of a × 2m power of B) = 5th power of a and 3rd power of B, try to find the value of M + n

(X-Y) seventh power ÷ [(X-Y) third power ÷ (Y-X)] third power
=(X-Y) seventh power ÷ [- (X-Y) ²】 Cubic
=(X-Y) seventh power ÷ [- (X-Y) sixth power]
=-(x-y)
=y-x
m+1+2n-1=5
n+2+2m=3
∴m=-1
n=3
∴m+n=2