The solutions for [tex]\log x + \log (3\cdot x - 13) = 1[/tex] are [tex]x_{1} = 5[/tex] and [tex]x_{2} = -\frac{2}{3}[/tex], respectively.
In this question, we are going to solve for [tex]x[/tex] with the help of Logarithm Properties, which are described in the image attached below.
[tex]\log x + \log (3\cdot x - 13) = 1[/tex]
[tex]\log [x\cdot (3\cdot x - 13)] = 1[/tex]
[tex]\log (3\cdot x^{2}-13\cdot x) = 1[/tex]
[tex]10^{\log(3\cdot x^{2}-13\cdot x)} = 10^{1}[/tex]
[tex]3\cdot x^{2}-13\cdot x = 10[/tex]
[tex]3\cdot x^{2}-13\cdot x -10 = 0[/tex]
This is a Second Order Polynomial, which can be solved by Quadratic Formula:
[tex]x_{1} = 5[/tex] and [tex]x_{2} = -\frac{2}{3}[/tex]
The solutions for [tex]\log x + \log (3\cdot x - 13) = 1[/tex] are [tex]x_{1} = 5[/tex] and [tex]x_{2} = -\frac{2}{3}[/tex], respectively.
Please see this question related to Logarithm Properties for further details:
https://brainly.com/question/12983107
