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 (1)
Solve the equation x − 4 = 2 3 to find x = 12 .
Substitute x = 12 into each of the logarithmic equations.
Check if the equation holds true for x = 12 .
The equation lo g 2 ( x − 4 ) = 3 has the same solution x = 12 , so the answer is lo g 2 ( x − 4 ) = 3 .
Explanation
Analyze the problem We are given the equation x − 4 = 2 3 and asked to find an equivalent logarithmic equation from the given options. First, let's simplify the given equation.
Solve for x We simplify x − 4 = 2 3 to x − 4 = 8 . Adding 4 to both sides, we get x = 12 . Now we need to find which of the given logarithmic equations also has the solution x = 12 .
Check option 1 Let's examine the first option: lo g 3 2 = ( x − 4 ) . This simplifies to lo g 9 = x − 4 . Substituting x = 12 , we get lo g 9 = 12 − 4 = 8 . This is false, since lo g 9 ≈ 2.197 = 8 .
Check option 2 Now let's examine the second option: lo g 2 3 = ( x − 4 ) . This simplifies to lo g 8 = x − 4 . Substituting x = 12 , we get lo g 8 = 12 − 4 = 8 . This is false, since lo g 8 ≈ 2.079 = 8 .
Check option 3 Now let's examine the third option: lo g 2 ( x − 4 ) = 3 . Substituting x = 12 , we get lo g 2 ( 12 − 4 ) = 3 , which simplifies to lo g 2 ( 8 ) = 3 . Since 2 3 = 8 , this is true. Therefore, this is the correct option.
Check option 4 Finally, let's examine the fourth option: lo g 3 ( x − 4 ) = 2 . Substituting x = 12 , we get lo g 3 ( 12 − 4 ) = 2 , which simplifies to lo g 3 ( 8 ) = 2 . This is false, since 3 2 = 9 = 8 .
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 instance, in computer science, logarithms are used to analyze the time complexity of algorithms. If an algorithm's time complexity is O ( lo g n ) , it means the algorithm's runtime increases logarithmically with the input size n . This is highly efficient, as the runtime grows very slowly as the input size increases. Understanding logarithmic equations helps in assessing and comparing the efficiency of different algorithms.
