Respuesta :
The true solution to the logarithmic equation log(x)+log(x+5)=log(6x+12) is 4 option second is correct.
It is given the logarithmic equation [tex]\rm logx+log(x+5)=log(6x+12)[/tex].
It is required to find the true solution to the above logarithmic equation.
What is Logarithm?
It is another way to represent the power of numbers and we say that 'b' is the logarithm of 'c' with base 'a' if and only if 'a' to the power 'b' equals 'c'.
[tex]\rm a^b=c\\[/tex]
[tex]\rm log_ac=b[/tex]
We have:
[tex]\rm logx+log(x+5)=log(6x+12)[/tex]
By using log property:
[tex]\rm logx + logy= log(xy)[/tex]
[tex]\rm log((x)(x+5))=log(6x+12)[/tex]
Taking [tex]\rm log_1_0[/tex] base and removing the log from the sides we get:
[tex]\rm x(x+5)= 6x+12\\\\\rm x^2+5x=6x+12\\\\\rm x^2-x-12=0[/tex]
It is quadratic equation solving by factorization method:
[tex]\rm x^2-x-12=0\\\\\rm x^2+3x-4x-12=0\\\\\rm x(x+3)-4(x+3)=0\\\\\rm (x+3)(x-4)=0\\\\\rm x+3=0 \ or \ x-4=0\\\\ \rm x=-3 \ or \ x=4\\\\[/tex]
A negative Logarithm value does not exist.
Hence, the true solution to the logarithmic equation log(x)+log(x+5)=log(6x+12) is x = 4
Learn more about the logarithm here:
https://brainly.com/question/163125
