Which logarithmic equation has the same solution as $x-4=2^3$?
A. $\log 3^2=(x-4)$
B. $\log 2^3=(x-4)$
C. $\log _2(x-4)=3$
D. $\log _3(x-4)=2$
Answer (2)
Solve the equation x − 4 = 2 3 to find x = 12 .
Substitute x = 12 into each logarithmic equation.
Check if the equation holds true with x = 12 .
The equation lo g 2 ( x − 4 ) = 3 holds true, since lo g 2 ( 12 − 4 ) = lo g 2 ( 8 ) = 3 .
The logarithmic equation with the same solution is lo g 2 ( x − 4 ) = 3 .
Explanation
Understanding the Problem We are given the equation x − 4 = 2 3 and asked to find an equivalent logarithmic equation from the given options. The options are:
lo g 3 2 = ( x − 4 )
lo g 2 3 = ( x − 4 )
lo g 2 ( x − 4 ) = 3
lo g 3 ( x − 4 ) = 2
Solving for x First, let's solve the given equation for x :
x − 4 = 2 3
Since 2 3 = 8 , the equation becomes:
x − 4 = 8
Adding 4 to both sides, we get:
x = 8 + 4 = 12
Checking the Logarithmic Equations Now, let's examine each of the logarithmic equations to see which one has the same solution, x = 12 .
lo g 3 2 = ( x − 4 )
Substituting x = 12 , we get lo g 3 2 = 12 − 4 = 8 . Since lo g 3 2 = lo g 9 ≈ 0.954 = 8 , this equation does not have the same solution.
lo g 2 3 = ( x − 4 )
Substituting x = 12 , we get lo g 2 3 = 12 − 4 = 8 . Since lo g 2 3 = lo g 8 ≈ 0.903 = 8 , this equation does not have the same solution.
lo g 2 ( x − 4 ) = 3
Substituting x = 12 , we get lo g 2 ( 12 − 4 ) = lo g 2 ( 8 ) = 3 . Since 2 3 = 8 , this equation is equivalent to x − 4 = 2 3 , and thus has the same solution.
lo g 3 ( x − 4 ) = 2
Substituting x = 12 , we get lo g 3 ( 12 − 4 ) = lo g 3 ( 8 ) = 2 . Since 3 2 = 9 = 8 , this equation does not have the same solution.
Final Answer Therefore, the logarithmic equation that has the same solution as x − 4 = 2 3 is lo g 2 ( x − 4 ) = 3 .
Examples
Logarithmic equations are used in various fields such as computer science, finance, and physics. For example, in computer science, logarithms are used to analyze the time complexity of algorithms. In finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. In physics, they appear in formulas related to entropy and information theory. Understanding how to manipulate and solve logarithmic equations is crucial for solving problems in these areas.
The equation given, x − 4 = 2 3 , solves to x = 12 . The logarithmic equation that has the same solution is log 2 ( x − 4 ) = 3 . Therefore, the correct answer is log 2 ( x − 4 ) = 3 .
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