Đáp án D.

Ta có

log   6125 7 = log   6125 + log 7 = log 7 2 . 125 + 1 2 log   7

= 5 2 log   7 + log   5 3 = 5 2 n + 3 log   5 = 5 2 n + 3 1 - log   2

= 5 2 n + 3 - 3 m .

\(log\left(5\left(x^2+1\right)\right)\ge log\left(mx^2+4x+m\right)\)

- BPT đúng \(\forall x\Rightarrow log\left(mx^2+4x+m\right)\) xác định \(\forall x\in R\)

\(\Rightarrow mx^2+4x+m>0\) \(\forall x\in R\)

\(\Rightarrow\left\{{}\begin{matrix}a=m>0\\\Delta'=4-m^2< 0\end{matrix}\right.\) \(\Rightarrow m>2\) (1)

- Lại có \(x^2+1\ge1\) \(\forall x\)

\(\Rightarrow5\left(x^2+1\right)\ge mx^2+4x+m\)

\(\Leftrightarrow5\left(x^2+1\right)-4x\ge m\left(x^2+1\right)\)

\(\Leftrightarrow5-\dfrac{4x}{x^2+1}\ge m\)

Đặt \(f\left(x\right)=5-\dfrac{4x}{x^2+1}\Rightarrow f\left(x\right)\ge m\) \(\forall x\Leftrightarrow m\le min\left(f\left(x\right)\right)\)

Ta có \(f\left(x\right)=3+2-\dfrac{4x}{x^2+1}=3+\dfrac{2\left(x-1\right)^2}{x^2+1}\ge3\)

\(\Rightarrow min\left(f\left(x\right)\right)=3\Rightarrow m\le3\) (2)

Kết hợp (1), (2) \(\Rightarrow2< m\le3\Rightarrow m=3\)

Vậy có 1 giá trị nguyên duy nhất của m để BPT đúng với mọi x

Đáp án B

a: \(log_2\left(M\cdot N\right)=log_2\left(2^5\cdot2^3\right)=log_2\left(2^8\right)=8\)

\(log_2M+log_2N=log_22^5+log_22^3=5+3=8\)

=>\(log_2\left(MN\right)=log_2M+log_2N\)

b: \(log_2\left(\dfrac{M}{N}\right)=log_2\left(\dfrac{2^5}{2^3}\right)=log_2\left(2^2\right)=2\)

\(log_2M-log_2N=log_22^5-log_22^3=5-3=2\)

=>\(log_2\left(\dfrac{M}{N}\right)=log_2M-log_2N\)

a: \(log_2\left(mn\right)=log_2\left(2^7\cdot2^3\right)=7+3=10\)

 \(log_2m+log_2n=log_22^7+log_22^3=7+3=10\)

=>\(log_2\left(mn\right)=log_2m+log_2n\)

b: \(log_2\left(\dfrac{m}{n}\right)=log_2\left(\dfrac{2^7}{2^3}\right)=7-3=4\)

\(log_2m-log_2n=log_22^7-log_22^3=7-3=4\)

=>\(log_2\left(\dfrac{m}{n}\right)=log_2m-log_2n\)

a) \(\log_2\left(mn\right)=\log_2\left(2^7.2^3\right)=\log_22^{7+3}=\log_22^{10}=10.\log_22=10.1=10\)

\(\log_2m+\log_2n=\log_22^7+\log_22^3=7\log_22+3\log_22=7.1+3.1=7+3=10\)

b) \(\log_2\left(\dfrac{m}{n}\right)=\log_2\dfrac{2^7}{2^3}=\log_22^4=4.\log_22=4.1=4\)

\(\log_2m-\log_2n=\log_22^7-\log_22^3=7.\log_22-3\log_22=7.1-3.1=4\)

\(log_xy=log_yx=\frac{1}{log_xy}\Rightarrow\left(log_xy\right)^2=1\Rightarrow\left[{}\begin{matrix}log_xy=1\\log_xy=-1\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=y\\x=\frac{1}{y}\end{matrix}\right.\)

Do \(log_x\left(x-y\right)\) tồn tại \(\Rightarrow x-y\ne0\Rightarrow x\ne y\Rightarrow x=\frac{1}{y}\)

\(log_x\left(x-y\right)=log_y\left(x+1\right)\Leftrightarrow log_x\left(x-\frac{1}{x}\right)=-log_x\left(x+1\right)\)

\(\Leftrightarrow log_x\left[\left(x-\frac{1}{x}\right)\left(x+1\right)\right]=0\Leftrightarrow\left(x-\frac{1}{x}\right)\left(x+1\right)=1\)

\(\Leftrightarrow\left(x^2-1\right)\left(x+1\right)=x\Leftrightarrow x^3+x^2-2x-1=0\)

Pt này nghiệm xấu, đề bài có vấn đề