Which of the following are true statements?
Check all that apply.
A. [tex]$\log M-\log N=\log (M-N)$[/tex]
B. [tex]$\log \left(\frac{M}{N}\right)=\log M-\log N$[/tex]
C. [tex]$\log M+\log N=\log (M N)$[/tex]
D. [tex]$\log \left(\frac{M}{N}\right)=\frac{\log M}{\log N}$[/tex]
Answer (2)
Recall the quotient rule: lo g ( N M ) = lo g M − lo g N .
Recall the product rule: lo g ( MN ) = lo g M + lo g N .
Statement A is false, while statements B and C are true.
Statement D is false. Therefore, the true statements are B and C.
Explanation
Understanding the Problem We are given four statements involving logarithms and asked to identify the true ones. The statements are: A. lo g M − lo g N = lo g ( M − N ) B. lo g ( N M ) = lo g M − lo g N C. lo g M + lo g N = lo g ( MN ) D. lo g ( N M ) = l o g N l o g M
Recalling Logarithmic Properties Let's recall the fundamental properties of logarithms that will help us evaluate these statements.
Product Rule: The logarithm of the product of two numbers is the sum of their logarithms: lo g ( MN ) = lo g M + lo g N .
Quotient Rule: The logarithm of the quotient of two numbers is the difference of their logarithms: lo g ( N M ) = lo g M − lo g N .
Power Rule: The logarithm of a number raised to a power is the product of the power and the logarithm of the number: lo g ( M p ) = p lo g M .
Evaluating the Statements Now, let's analyze each statement:
Statement A: lo g M − lo g N = lo g ( M − N ) . This statement is false . The correct identity is lo g M − lo g N = lo g ( N M ) .
Statement B: lo g ( N M ) = lo g M − lo g N . This statement is true , as it directly represents the quotient rule of logarithms.
Statement C: lo g M + lo g N = lo g ( MN ) . This statement is true , as it directly represents the product rule of logarithms.
Statement D: lo g ( N M ) = l o g N l o g M . This statement is false . The correct identity is lo g ( N M ) = lo g M − lo g N , not l o g N l o g M .
Conclusion Therefore, statements B and C are the true statements.
Examples
Logarithms are incredibly useful in many real-world applications. For example, in chemistry, the pH scale uses logarithms to measure the acidity or alkalinity of a solution. In finance, logarithms are used to calculate compound interest and analyze investment growth. In seismology, the Richter scale uses logarithms to measure the magnitude of earthquakes. Understanding logarithmic properties allows scientists and professionals to simplify complex calculations and make informed decisions in various fields.
The true statements are B: lo g ( N M ) = lo g M − lo g N and C: lo g M + lo g N = lo g ( MN ) . Statements A and D are false. These results are based on the logarithmic properties of the quotient and product rules.
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