ANSWER
x = 4 or x = -9
STEP-BY-STEP EXPLANATION:
As you can see from the question provided, you were given the following logarithms equation
[tex]\log _6x+log_6(x\text{ + 5) = 2}[/tex]
Recall that,
[tex]\text{Log}_xA+log_xB=Log_x(A\text{ x B)}[/tex][tex]\begin{gathered} \text{Let} \\ \log _xA=log_6x \\ \log _xB=log_6(x\text{ + 5)} \end{gathered}[/tex]
The next thing is to replace the values and simplify
[tex]\begin{gathered} \log _6x(x\text{ + 5) = 2} \\ x(x+5)=6^2 \\ \text{Open the parenthesis} \\ x^2\text{ + 5x = 36} \\ x^2\text{ + 5x - 36 = 0} \end{gathered}[/tex]
As you can see from the last process, the logarithms function resulted in a quadratic equation.
The next thing is to solve the equation using the factorization method.
Note that, the general quadratic function is given below as
[tex]ax^2\text{ + bx + c = 0}[/tex]
Relating the two equations, you will have the following data
• a = 1
,
• b = 5
,
• c = -36
The next thing is to find the value of ac
[tex]\begin{gathered} ac\text{ = 1 }\cdot\text{(-36)} \\ ac\text{ = -36} \end{gathered}[/tex][tex]\begin{gathered} x^2\text{ -4x + 9x - 36 = 0} \\ x(x\text{ - 4) + 9(x - 4) = 0} \\ (x\text{ - 4) (x + 9) = 0} \\ (x\text{ - 4) = 0 or (x + 9) = 0} \\ x\text{ - 4 = 0 or x + 9 = 0} \\ x\text{ = 0 + 4 or x - 9 = 0} \\ x\text{ = 4 or x =-9} \end{gathered}[/tex]
Hence, the values of x are 4 and -9
