Logarithmus Brüche umformen lg(2x+3)=lg(x-1)+1

Question

Wie löse ich diese Gleichung nach x auf?

lg(2x+3)=lg(x-1)+1

Answer 1

\(\lg (2 x+3)=\lg (x-1)+1 \quad |- lg(x-1)\)
\(\lg (2 x+3)-\lg (x-1)=1\)
\(\lg \left(\frac{2 x+3}{x-1}\right)=1 \quad | 10^{()}\)
\(\frac{2 x+3}{x-1}=10^{1}=10\)
\(\begin{aligned} 2 x+3 &=10(x-1) \\ 2 x+3 &=10 x-10 \quad |-3 \\ 2 x &=10 x-13 \quad |-10 x \\-8 x &=-13 \end{aligned}\)
\(x=\frac{13}{8}\)

Answer 2

$$ lg(2x+3)=lg(x-1)+1 \\lg(2x+3)=lg(x-1)+lg(10)\\ lg(2x+3)=lg(10x-10)\\2x+3=10x-10\\13=8x\\x=13/8 $$