Równania logarytmiczne

[hojrak]

a) \(\log_\frac{1}{3}(x^2+2x+3)=-1\)
b) \(\log_2(x^2-1)=3\)
c) \(\log_{\sqrt{5}}(9-4x^2)=2\)
d) \(\log_{0,5}(x^2+4x+4)=-2\)
e) \(\log_{27}(x^2-4x+3)=\frac{1}{3}\)
f) \(\log_{2\sqrt{2}}(5+4x-x^2)=2\)

[eresh]

a) \(\log_\frac{1}{3}(x^2+2x+3)=-1\)

\(x^2+2x+3>0\\
x\in\mathbb{R}\\
D=\mathbb{R}\)

\(x^2+2x+3=(\frac{1}{3})^{-1}\\
x^2+2x+3=3\\
x^2+2x=0\\
x(x+2)=0\\
x=0\;\;\vee\;\;x=-2\)

[eresh]

\(\)

b) \(\log_2(x^2-1)=3\)

\(x^2-1>0\\
D=(-\infty, -1)\cup (1,\infty)\)

\(x^2-1=2^3\\
x^2-1=8\\
x^2=9\\
x=3\\
x=-3\)

[eresh]

c) \(\log_{\sqrt{5}}(9-4x^2)=2\)

\(9-4x^2>0\\
(3-2x)(3+2x)>0\\
D=(-\frac{3}{2},\frac{3}{2})\)

\(9-4x^2=\sqrt{5}^2\\
9-4x^2=5\\
4x^2=4\\
x^2=1\\
x=-1\;\;\vee\;\;x=1\)

[eresh]

d) \(\log_{0,5}(x^2+4x+4)=-2\)

\(x^2+4x+4>0\\
D=\mathbb{R}\setminus\{-2\}\)

\(x^2+4x+4=4\\
x(x+4)=0\\
x=0\;\;\vee\;\;x=-4\)

[eresh]

e) \(\log_{27}(x^2-4x+3)=\frac{1}{3}\)

\(x^2-4x+3>0\\
(x-1)(x-3)>0\\
D=(-\infty, 1)\cup (3,\infty)\)

\(x^2-4x+3=3\\
x(x-4)=0\\
x=0\;\;x=4\)

[eresh]

f) \(\log_{2\sqrt{2}}(5+4x-x^2)=2\)

\(5+4x-x^2>0\\
-(x+1)(x-5)>0\\
D=(-1,5)\)

\(5+4x-x^2=8\\
-x^2+4x-3=0\\
x=1\;\;\vee\;\;x=3\)