Solve the equation.
[tex]\begin{array}{c}\log _6(x-2)+\log _6(x+3)=2 \\x=[?]\end{array}[/tex]
Answer (2)
Combine the logarithms using the property lo g a ( b ) + lo g a ( c ) = lo g a ( b c ) .
Convert the logarithmic equation to an exponential equation.
Solve the resulting quadratic equation by factoring.
Check the solutions against the domain of the logarithmic functions to eliminate extraneous solutions. The final answer is 6 .
Explanation
Domain Analysis We are given the equation lo g 6 ( x − 2 ) + lo g 6 ( x + 3 ) = 2 and we need to solve for x . First, we need to consider the domain of the logarithmic functions. For lo g 6 ( x − 2 ) to be defined, we need 0"> x − 2 > 0 , which means 2"> x > 2 . Similarly, for lo g 6 ( x + 3 ) to be defined, we need 0"> x + 3 > 0 , which means -3"> x > − 3 . Combining these two conditions, we must have 2"> x > 2 .
Combining Logarithms Using the property of logarithms that lo g a ( b ) + lo g a ( c ) = lo g a ( b c ) , we can rewrite the given equation as lo g 6 (( x − 2 ) ( x + 3 )) = 2.
Converting to Exponential Form Now, we convert the logarithmic equation to an exponential equation: ( x − 2 ) ( x + 3 ) = 6 2 = 36.
Expanding the Equation Expanding the left side of the equation, we get x 2 + 3 x − 2 x − 6 = 36 , which simplifies to x 2 + x − 6 = 36.
Simplifying to Quadratic Form Subtracting 36 from both sides, we obtain the quadratic equation x 2 + x − 42 = 0.
Factoring the Quadratic We can factor the quadratic equation as ( x − 6 ) ( x + 7 ) = 0.
Checking for Extraneous Solutions This gives us two possible solutions for x : x = 6 or x = − 7 . However, we must check these solutions against the domain restriction 2"> x > 2 . Since − 7 ≯ 2 , x = − 7 is not a valid solution. The other solution, x = 6 , satisfies the condition 2"> x > 2 . Therefore, the only solution is x = 6 .
Final Answer Thus, the solution to the equation lo g 6 ( x − 2 ) + lo g 6 ( x + 3 ) = 2 is x = 6 .
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. Understanding how to solve logarithmic equations is crucial for making accurate predictions and analyses in these areas. For example, if we know the intensity of an earthquake, we can use a logarithmic equation to find its magnitude on the Richter scale, which helps in assessing the potential damage and impact on surrounding areas.
The solution to the equation lo g 6 ( x − 2 ) + lo g 6 ( x + 3 ) = 2 is found to be x = 6 . After analyzing the domain and combining the logarithmic expressions, we created a quadratic equation, which we solved to find the correct answer. The extraneous solution x = − 7 was eliminated based on the domain restrictions.
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