\section*{Problem 4 - 141234}

It is to be investigated whether there is a function \(y=\log _{a}(b x+c)\) with \(a, b, c\) real; \(a>1\) whose graph in an \(x, y\)-coordinate system passes through the points \((2 ; 2),(-1 ; 0)\) and \((0 ; 1)\).

Give, if such a function exists, all real ordered triples \((a, b, c)\) for which this is true.
}

Due to \(1=y(0)=\log _{a}(c)\), it follows that \(c=a\). Due to

\[
0=y(-1)=\log _{a}((1-) \cdot b+c)=\log _{a}(c-b)
\]

it follows that \(c-b=1\), so \(b=c-1=a-1\). And due to

\[
2=y(2)=\log _{a}(2 b+c)=\log _{a}(3 a-2)
\]

it follows that \(a^{2}=3 a-2\), or \(a^{2}-3 a+2=0\) respectively,

\[
a=\frac{3}{2} \pm \sqrt{\frac{9}{4}-2}=\frac{3 \pm 1}{2}
\]

where the solution of the quadratic equation with the negative sign in front of the square root is discarded, as it leads to \(a=1\), which contradicts the problem statement. Therefore, \(a=c=2\) and \(b=1\) must hold. Substituting these values confirms that the corresponding function passes through the three points, so there is exactly one such function, namely the one with the parameters \((a, b, c)=(2,1,2)\).