Which logarithmic equation has the same solution as [tex]2^{3} = (x - 4)[/tex]?
Answer (1)
Solve the equation 2 x − 4 = 2 3 to find x = 6 .
Substitute x = 6 into each of the given logarithmic equations to see if it satisfies the equation.
Solve each of the logarithmic equations for x and compare the solution to x = 6 .
Conclude that none of the given logarithmic equations have the same solution as the equation 2 x − 4 = 2 3 .
Explanation
Solving the Linear Equation We are given the equation 2 x − 4 = 2 3 and asked to find which of the given logarithmic equations has the same solution. Let's first solve the given equation for x .
Simplifying the Equation First, we simplify 2 3 to 8, so the equation becomes 2 x − 4 = 8 .
Isolating the Term with x Next, we add 4 to both sides of the equation: 2 x = 8 + 4 , which simplifies to 2 x = 12 .
Finding the Value of x Finally, we divide both sides by 2 to solve for x : x = 2 12 = 6 . So, the solution to the given equation is x = 6 .
Checking the Logarithmic Equations Now, let's examine the given logarithmic equations to see which one has the same solution, x = 6 .
lo g 3 ( x − 4 ) = 2 . Substituting x = 6 , we get lo g 3 ( 6 − 4 ) = lo g 3 ( 2 ) . Since lo g 3 ( 2 ) = 2 , this equation does not have the same solution.
3 2 = ( x − 4 ) . Substituting x = 6 , we get 3 2 = 6 − 4 , which simplifies to 9 = 2 . This is false, so this equation does not have the same solution.
2 3 = ( x − 4 ) . Substituting x = 6 , we get 2 3 = 6 − 4 , which simplifies to 8 = 2 . This is false, so this equation does not have the same solution.
lo g 2 ( x − 4 ) = 3 . Substituting x = 6 , we get lo g 2 ( 6 − 4 ) = lo g 2 ( 2 ) . Since lo g 2 ( 2 ) = 1 = 3 , this equation does not have the same solution.
lo g 3 ( x − 4 ) = 2 . Substituting x = 6 , we get lo g 3 ( 6 − 4 ) = lo g 3 ( 2 ) . Since lo g 3 ( 2 ) = 2 , this equation does not have the same solution.
However, let's solve each of the logarithmic equations for x :
lo g 3 ( x − 4 ) = 2 ⟹ x − 4 = 3 2 = 9 ⟹ x = 13
2 3 = x − 4 ⟹ 8 = x − 4 ⟹ x = 12
lo g 2 ( x − 4 ) = 3 ⟹ x − 4 = 2 3 = 8 ⟹ x = 12
lo g 3 ( x − 4 ) = 2 ⟹ x − 4 = 3 2 = 9 ⟹ x = 13
Final Analysis Upon re-evaluating the options, we see that option 3, presented as lo g 2 ( x − 4 ) = 3 , when solved for x, gives us x = 12 . The equation 2 3 = x − 4 also gives x = 12 . However, the original equation 2 x − 4 = 2 3 gives x = 6 . Thus, none of the given options have the same solution as the original equation. There seems to be a typo in the original options. Let's consider the equation lo g 2 ( x − 4 ) = 3 . Solving this, we have x − 4 = 2 3 = 8 , so x = 12 . The original equation is 2 x − 4 = 2 3 = 8 , so 2 x = 12 and x = 6 . None of the equations have the same solution.
Checking Equivalence However, if the question meant to ask which equation is equivalent to 2 x − 4 = 2 3 , we solve for x to get 2 x − 4 = 8 , so 2 x = 12 and x = 6 . Substituting x = 6 into the equations:
lo g 3 ( x − 4 ) = 2 gives lo g 3 ( 6 − 4 ) = lo g 3 ( 2 ) = 2
3 2 = x − 4 gives 9 = 6 − 4 = 2 , which is false.
2 3 = x − 4 gives 8 = 6 − 4 = 2 , which is false.
lo g 2 ( x − 4 ) = 3 gives lo g 2 ( 6 − 4 ) = lo g 2 ( 2 ) = 1 = 3
None of the equations are equivalent to the original equation.
Further Analysis Let's assume the original equation was meant to be 2 x − 4 = 4 , then 2 x = 8 and x = 4 . Then:
lo g 3 ( x − 4 ) = 2 gives lo g 3 ( 4 − 4 ) = lo g 3 ( 0 ) , which is undefined.
3 2 = x − 4 gives 9 = 4 − 4 = 0 , which is false.
2 3 = x − 4 gives 8 = 4 − 4 = 0 , which is false.
lo g 2 ( x − 4 ) = 3 gives lo g 2 ( 4 − 4 ) = lo g 2 ( 0 ) , which is undefined.
If the original equation was meant to be x = 6 , then we want to find which equation gives x = 6 .
lo g 3 ( x − 4 ) = 2 gives x − 4 = 3 2 = 9 , so x = 13
2 3 = x − 4 gives 8 = x − 4 , so x = 12
lo g 2 ( x − 4 ) = 3 gives x − 4 = 2 3 = 8 , so x = 12
lo g 3 ( x − 4 ) = 2 gives x − 4 = 3 2 = 9 , so x = 13
None of the equations give x = 6 .
Conclusion Based on the analysis, none of the provided logarithmic equations have the same solution as the equation 2 x − 4 = 2 3 .
Examples
Logarithmic equations are used in many fields, including computer science, finance, and physics. For example, in computer science, logarithms are used to measure the time complexity of algorithms. In finance, logarithms are used to calculate compound interest. In physics, logarithms are used to measure the intensity of sound and light. Understanding how to solve logarithmic equations is therefore essential for anyone working in these fields.
