If [tex]$\log \frac{M}{N}=4$[/tex] and [tex]$\log \frac{P}{N}=5$[/tex], what can you say about the relationship between [tex]$M$[/tex] and [tex]$P$[/tex]?
A. [tex]$P=0.1 M$[/tex]
B. [tex]$P=100 M$[/tex]
C. [tex]$M=10 P$[/tex]
D. [tex]$P=10 M$[/tex]
Answer (2)
Rewrite the logarithmic equations as exponential equations: N M = 1 0 4 and N P = 1 0 5 .
Solve for N in both equations: N = 1 0 4 M and N = 1 0 5 P .
Equate the two expressions for N : 1 0 4 M = 1 0 5 P .
Solve for P in terms of M : P = 10 M . The answer is P = 10 M .
Explanation
Understanding the Problem We are given two logarithmic equations: lo g N M = 4 and lo g N P = 5 . Our goal is to find the relationship between M and P . Since the base of the logarithm is not specified, we assume it is base 10.
Rewriting the Equations We can rewrite the given equations using the definition of logarithms. The equation lo g N M = 4 implies that N M = 1 0 4 . Similarly, the equation lo g N P = 5 implies that N P = 1 0 5 .
Solving for N Now, we solve for N in both equations. From the first equation, we have N = 1 0 4 M . From the second equation, we have N = 1 0 5 P .
Equating the Expressions for N Since both expressions are equal to N , we can set them equal to each other: 1 0 4 M = 1 0 5 P .
Solving for P To find the relationship between M and P , we can solve for P in terms of M . Multiplying both sides of the equation by 1 0 5 , we get: P = 1 0 4 1 0 5 M = 10 M .
Final Answer Therefore, the relationship between M and P is P = 10 M . This corresponds to option D.
Examples
Logarithms are used in many real-world applications, such as measuring the magnitude of earthquakes on the Richter scale or the loudness of sound in decibels. Understanding logarithmic relationships helps scientists and engineers analyze and interpret data in these fields. For example, if the magnitude of one earthquake is 4 and another is 5, the second earthquake is actually 10 times stronger than the first, demonstrating the relationship we found between M and P.
The relationship between M and P is defined by the equation P = 10 M, which corresponds to option D. This indicates that P is ten times the value of M. Therefore, if M increases, P will also increase proportionally by a factor of ten.
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