Mister Exam
2^2+log2^(x^2+y^2)=20; Lg(x^2-y^2)-lg(x-y)=0
The solution
You have entered
[src]
2 2
x + y
4 + (log(2)) = 20
$$\log{\left(2 \right)}^{x^{2} + y^{2}} + 4 = 20$$
/ 2 2\ log\x - y / - log(x - y) = 0
$$- \log{\left(x - y \right)} + \log{\left(x^{2} - y^{2} \right)} = 0$$
-log(x - y) + log(x^2 - y^2) = 0
Rapid solution
$$x_{1} = \frac{1}{2} - \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
$$\frac{1}{2} - \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
$$y_{1} = \frac{1}{2} + \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
$$\frac{1}{2} + \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
=
$$\frac{1}{2} - \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
0.5 + 2.00808087164282*i
$$y_{1} = \frac{1}{2} + \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
$$\frac{1}{2} + \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
0.5 - 2.00808087164282*i
$$x_{2} = \frac{1}{2} + \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
$$\frac{1}{2} + \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
$$y_{2} = \frac{1}{2} - \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
$$\frac{1}{2} - \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
=
$$\frac{1}{2} + \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
0.5 - 2.00808087164282*i
$$y_{2} = \frac{1}{2} - \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
$$\frac{1}{2} - \frac{\sqrt{\log{\left(\frac{256}{\log{\left(2 \right)}} \right)}}}{2 \sqrt{\log{\left(\log{\left(2 \right)} \right)}}}$$
=
0.5 + 2.00808087164282*i