Answer :
Logarithms are a way to find how much a number must be raised in order to get the desired number. The solutions to[tex]\rm log_6(x^2+8)= 1+log_6(x)[/tex] are 2 and 4.
What are the Properties of logarithms?
There are four basic properties of logarithms:
[tex]\rm log_aU+ log_aV = log_a(UV)\\\\\rm log_aU - log_aV = log_a(\dfrac{U}{V})\\\\\rm log_aU^n = n\ log_aU\\\\\rm log_ab = \dfrac{log_xb}{log_xa}[/tex]
In order to solve the problem, we will use some of the basic logarithmic properties, therefore, the solution of the problem is,
[tex]\rm log_6(x^2+8)= 1+log_6(x)\\\\-1= log_6(x)+ log_6(x^2+8)\\\\\text{Using the property we get}\\\\-1 = log_6\dfrac{x}{(x^2+8)}\\[/tex]
Taking antilog, we get,
[tex]6^{-1} = \dfrac{x}{(x^2+8)}\\\\(x^2+8)= \dfrac{x}{6^{-1} }\\\\x^2 + 8 = 6x\\\\x^2 -6x+8=0\\[/tex]
Solving the quadratic equation,
[tex]x^2-6x+8=0\\\\x^2 -4x-2x+8=0\\\\x(x-4)-2(x-4)=0\\\\(x-2)(x-4)=0[/tex]
Substituting the roots with zero we get,
[tex]x-2 = 0\\x =2\\\\x -4 = 0\\x = 4[/tex]
hence, the solutions to[tex]\rm log_6(x^2+8)= 1+log_6(x)[/tex] are 2 and 4.
Learn more about Logarithms:
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