Solve for $x$.
$\log _2(x-3)=2-\log _2(x-6)$
If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".
$x=$
Answer (2)
Rewrite the equation using logarithm properties: lo g 2 (( x − 3 ) ( x − 6 )) = 2 .
Remove the logarithm by exponentiating: ( x − 3 ) ( x − 6 ) = 4 .
Simplify to a quadratic equation: x 2 − 9 x + 14 = 0 .
Solve the quadratic equation by factoring: ( x − 2 ) ( x − 7 ) = 0 , which gives x = 2 and x = 7 . Check the domain 6"> x > 6 . The final answer is 7 .
Explanation
Analyze the problem and domain We are given the equation lo g 2 ( x − 3 ) = 2 − lo g 2 ( x − 6 ) . Our goal is to solve for x . First, we need to consider the domain of the logarithmic functions. We must have 0"> x − 3 > 0 and 0"> x − 6 > 0 . This implies that 3"> x > 3 and 6"> x > 6 . Therefore, the solution must satisfy 6"> x > 6 .
Rewrite the equation We can rewrite the given equation by adding lo g 2 ( x − 6 ) to both sides: lo g 2 ( x − 3 ) + lo g 2 ( x − 6 ) = 2 Using the property of logarithms that lo g a ( b ) + lo g a ( c ) = lo g a ( b c ) , we have lo g 2 (( x − 3 ) ( x − 6 )) = 2
Remove the logarithm To remove the logarithm, we exponentiate both sides with base 2: 2 l o g 2 (( x − 3 ) ( x − 6 )) = 2 2 This simplifies to ( x − 3 ) ( x − 6 ) = 4
Expand and simplify Expanding the left side, we get x 2 − 6 x − 3 x + 18 = 4 x 2 − 9 x + 18 = 4 Subtracting 4 from both sides, we obtain the quadratic equation x 2 − 9 x + 14 = 0
Solve the quadratic equation We can solve this quadratic equation by factoring. We look for two numbers that multiply to 14 and add to -9. These numbers are -2 and -7. Thus, we can factor the quadratic as ( x − 2 ) ( x − 7 ) = 0 This gives us two possible solutions: x = 2 and x = 7 .
Check the solutions Now we need to check if these solutions satisfy the domain restriction 6"> x > 6 . The solution x = 2 does not satisfy this condition, so it is not a valid solution. The solution x = 7 does satisfy the condition 6"> x > 6 , so it is a valid solution.
Final Answer Therefore, the only solution is x = 7 .
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 or decay in biology. For instance, if we know the initial and final population sizes and the time elapsed, we can use logarithmic equations to find the growth rate. These equations are also crucial in finance for calculating compound interest and in computer science for analyzing the efficiency of algorithms.
The solution to the equation lo g 2 ( x − 3 ) = 2 − lo g 2 ( x − 6 ) is x = 7 . This solution is valid since it satisfies the logarithmic domain requirements. No other solutions meet the domain constraints, thus confirming x = 7 is the sole solution.
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