Which of the following pairs are equivalent statements?
A. [tex]$\log _b N=p$[/tex] and [tex]$b^p=N$[/tex]
B. [tex]$N^{1 / b}=p$[/tex] and [tex]$b^p=N$[/tex]
C. [tex]$\log _p N=b$[/tex] and [tex]$b^p=N$[/tex]
D. [tex]$\log _N b=p$[/tex] and [tex]$b^p=N$[/tex]
Answer (2)
The logarithmic equation lo g b N = p is equivalent to the exponential equation b p = N by definition.
The equation N 1/ b = p is equivalent to N = p b , which is not the same as b p = N .
The equation lo g p N = b is equivalent to p b = N , which is not the same as b p = N .
The equation lo g N b = p is equivalent to N p = b , which is not the same as b p = N . Therefore, only option A is equivalent: A .
Explanation
Understanding the Problem We are given four pairs of statements and need to determine which pairs are equivalent.
Listing the Pairs Pair A: lo g b N = p and b p = N Pair B: N 1/ b = p and b p = N Pair C: lo g p N = b and b p = N Pair D: lo g N b = p and b p = N
Objective We need to check each pair to see if the statements are equivalent.
Analyzing Pair A Pair A: The logarithmic equation lo g b N = p is equivalent to the exponential equation b p = N by the definition of logarithms. So, this pair is equivalent.
Analyzing Pair B Pair B: The equation N 1/ b = p can be rewritten as N = p b . This is NOT equivalent to b p = N . So, this pair is not equivalent.
Analyzing Pair C Pair C: The logarithmic equation lo g p N = b is equivalent to the exponential equation p b = N . This is NOT equivalent to b p = N . So, this pair is not equivalent.
Analyzing Pair D Pair D: The logarithmic equation lo g N b = p is equivalent to the exponential equation N p = b . This is NOT equivalent to b p = N . So, this pair is not equivalent.
Conclusion Therefore, only pair A consists of equivalent statements.
Examples
Logarithms and exponentials are used in many real-world applications, such as calculating the magnitude of earthquakes using the Richter scale, modeling population growth, and determining the decay of radioactive materials. For example, if you know the intensity of an earthquake, you can use logarithms to find its magnitude. Similarly, if you know the growth rate of a population, you can use exponentials to predict its future size. These concepts are fundamental in science and engineering.
Only Pair A, which states lo g b N = p and b p = N , consists of equivalent statements. All other pairs are not equivalent. Therefore, the correct answer is A.
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