Solve the equation: [tex]$\log _2(x)^0+\log _2(x-7)=3$[/tex]
The solution is [tex]$x = \square$[/tex]
Answer (2)
Simplify the equation using the property lo g 2 ( x ) 0 = 1 .
Isolate the logarithmic term: lo g 2 ( x − 7 ) = 2 .
Remove the logarithm by exponentiating: x − 7 = 2 2 = 4 .
Solve for x : x = 11 , which satisfies the domain restrictions. The final answer is 11 .
Explanation
Problem Analysis We are given the equation lo g 2 ( x ) 0 + lo g 2 ( x − 7 ) = 3 . Our goal is to solve for x .
Simplifying the Equation First, we simplify the equation. Recall that any non-zero number raised to the power of 0 is 1. Thus, lo g 2 ( x ) 0 = 1 , provided that 0"> x > 0 and x = 1 . The equation becomes 1 + lo g 2 ( x − 7 ) = 3 .
Isolating the Logarithm Next, we isolate the logarithmic term by subtracting 1 from both sides: lo g 2 ( x − 7 ) = 3 − 1 = 2 .
Removing the Logarithm To remove the logarithm, we exponentiate both sides with base 2: x − 7 = 2 2 = 4 .
Solving for x Now, we solve for x by adding 7 to both sides: x = 4 + 7 = 11 .
Checking the Solution We need to check if the solution is valid. First, we need 0"> x > 0 and x = 1 . Since x = 11 , this condition is satisfied. Also, the argument of the logarithm must be positive, so 0"> x − 7 > 0 , which means 7"> x > 7 . Since 7"> 11 > 7 , the solution is valid.
Final Answer Therefore, the solution to the equation is x = 11 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution, and in financial mathematics to calculate the time it takes for an investment to double at a certain interest rate. Understanding how to solve logarithmic equations helps in these practical applications.
To solve the equation lo g 2 ( x ) 0 + lo g 2 ( x − 7 ) = 3 , we simplify it to isolate the logarithmic term, leading to lo g 2 ( x − 7 ) = 2 . After exponentiating and solving, we find that x = 11 , which is a valid solution. Therefore, the final answer is x = 11 .
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