What are the solutions to $\log _6\left(x^2+8\right)=1+\log _6(x)$?
A. $x=-2$ and $x=-4$
B. $x=-1$ and $x=8$
C. $x=1$ and $x=-8$
D. $x=2$ and $x=4
Answer (2)
Rewrite the equation using logarithm properties: lo g 6 ( x 2 + 8 ) = lo g 6 ( 6 x ) .
Equate the arguments: x 2 + 8 = 6 x .
Rearrange to form a quadratic equation: x 2 − 6 x + 8 = 0 .
Factor and solve: ( x − 2 ) ( x − 4 ) = 0 , so x = 2 and x = 4 . The final answer is x = 2 and x = 4 .
Explanation
Understanding the Problem We are given the equation lo g 6 ( x 2 + 8 ) = 1 + lo g 6 ( x ) . We need to find the values of x that satisfy this equation. The domain of the logarithm function requires that 0"> x 2 + 8 > 0 and 0"> x > 0 . Since x 2 + 8 is always positive, we only need to consider 0"> x > 0 .
Rewriting the Equation We can rewrite the equation using logarithm properties. Since 1 = lo g 6 ( 6 ) , we have lo g 6 ( x 2 + 8 ) = lo g 6 ( 6 ) + lo g 6 ( x ) Using the property lo g a ( b ) + lo g a ( c ) = lo g a ( b c ) , we get lo g 6 ( x 2 + 8 ) = lo g 6 ( 6 x )
Forming a Quadratic Equation Since the logarithms are equal, we can equate the arguments: x 2 + 8 = 6 x Rearranging the equation, we get a quadratic equation: x 2 − 6 x + 8 = 0
Solving the Quadratic Equation We can factor the quadratic equation: ( x − 2 ) ( x − 4 ) = 0 Solving for x , we get: x = 2 or x = 4
Checking the Solutions We need to check if the solutions are valid. Since the domain restriction is 0"> x > 0 , both x = 2 and x = 4 satisfy this condition. Therefore, the solutions are x = 2 and x = 4 .
Final Answer The solutions to the equation lo g 6 ( x 2 + 8 ) = 1 + lo g 6 ( x ) are x = 2 and x = 4 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. In finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. Understanding how to solve logarithmic equations allows us to make predictions and analyze data in these real-world scenarios.
The solutions to the equation lo g 6 ( x 2 + 8 ) = 1 + lo g 6 ( x ) are x = 2 and x = 4 .
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