Let [tex]$\log \frac{P}{N}=6$[/tex] and [tex]$\log \frac{M}{N}=8$[/tex]. What is the relationship between [tex]$P$[/tex] and [tex]$M$[/tex]?
A. [tex]$M=100 P$[/tex]
B. [tex]$M=.001 P$[/tex]
C. [tex]$P=100 M$[/tex]
D. [tex]$M=2 P$[/tex]
Answer (2)
Rewrite the logarithmic equations as exponential equations: N P = 1 0 6 and N M = 1 0 8 .
Solve for N in both equations: N = 1 0 6 P and N = 1 0 8 M .
Set the two expressions for N equal to each other: 1 0 6 P = 1 0 8 M .
Solve for M in terms of P : M = 100 P , so the final answer is M = 100 P .
Explanation
Understanding the Problem We are given two logarithmic equations: lo g N P = 6 and lo g N M = 8 . Our goal is to find the relationship between P and M .
Rewriting the Equations Since the logarithm is not explicitly specified, we assume it is the base-10 logarithm. We can rewrite the given equations using the definition of logarithm: N P = 1 0 6 N M = 1 0 8
Solving for N Now, we solve for N in both equations. From the first equation, we have: N = 1 0 6 P From the second equation, we have: N = 1 0 8 M
Equating the Expressions for N Since both expressions are equal to N , we can set them equal to each other: 1 0 6 P = 1 0 8 M
Solving for M Now, we solve for M in terms of P . Multiply both sides by 1 0 8 : M = 1 0 6 1 0 8 P Simplify the fraction: M = 1 0 8 − 6 P M = 1 0 2 P M = 100 P
Conclusion Therefore, the relationship between P and M is M = 100 P .
Examples
Logarithms are used to measure the magnitude of earthquakes on the Richter scale. If the magnitude of one earthquake is 6 and another is 8, the difference in their magnitudes can be related to the ratio of their amplitudes. Similarly, in finance, logarithmic scales are used to compare investment returns, and understanding the relationship between these returns can help in making informed decisions. This problem demonstrates how logarithmic relationships can be transformed into direct relationships between variables, which is a common technique in many scientific and engineering applications.
The relationship between P and M is given by M = 100 P . This is derived from converting the logarithmic equations into exponential form and solving for M in terms of P.
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