What is $3 \ln 3-\ln 9$ expressed as a single natural logarithm?
A. $\ln 3$
B. $\ln 6$
C. $\ln 18$
D. $\ln 27$
Answer (2)
Use the power rule of logarithms to rewrite 3 ln 3 as ln 3 3 = ln 27 .
Rewrite the expression as ln 27 − ln 9 .
Use the quotient rule of logarithms to rewrite ln 27 − ln 9 as ln 9 27 .
Simplify the fraction 9 27 to get 3, so the final expression is ln 3 .
Explanation
Understanding the problem We are given the expression 3 ln 3 − ln 9 and asked to express it as a single natural logarithm. To do this, we will use properties of logarithms.
Applying the power rule First, we use the power rule of logarithms, which states that a ln x = ln x a . Applying this to the first term, we have
3 ln 3 = ln 3 3 = ln 27 .
Rewriting the expression Now, we can rewrite the original expression as
ln 27 − ln 9 .
Applying the quotient rule Next, we use the quotient rule of logarithms, which states that ln x − ln y = ln y x . Applying this to our expression, we have
ln 27 − ln 9 = ln 9 27 .
Simplifying the fraction Finally, we simplify the fraction:
9 27 = 3 .
Therefore, the expression simplifies to ln 3 .
Final Answer Thus, 3 ln 3 − ln 9 expressed as a single natural logarithm is ln 3 .
So the answer is ln 3 .
Examples
Logarithms are incredibly useful in many real-world situations. For example, they are used to measure the magnitude of earthquakes on the Richter scale. The formula is M = lo g 10 ( A ) , where M is the magnitude and A is the amplitude of the seismic waves. Logarithms also appear in finance when calculating compound interest and in chemistry when determining the pH of a solution. Understanding how to manipulate logarithmic expressions allows us to solve complex problems in various fields.
The expression 3 ln 3 − ln 9 simplifies to ln 3 . This is achieved by using the power and quotient rules of logarithms. Therefore, the correct answer is oxed{\text{A. } \text{ln} 3} .
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