What is $3 \ln 3-\ln 9$ expressed as a single natural logarithm?
A. In 3
B. In 6
C. In 18
D. In 27
Answer (2)
Use the power rule of logarithms to rewrite the expression: 3 ln 3 = ln 3 3 = ln 27 .
Use the quotient rule of logarithms to combine the terms: ln 27 − ln 9 = ln 9 27 .
Simplify the fraction: 9 27 = 3 .
Express the result as a single natural logarithm: ln 3 .
Explanation
Understanding the problem We are given the expression 3 ln 3 − ln 9 and we want 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.
Applying the quotient rule Now our expression is ln 27 − ln 9 . We use the quotient rule of logarithms, which states that ln x − ln y = ln y x . Applying this rule, we get ln 27 − ln 9 = ln 9 27 .
Simplifying the fraction Finally, we simplify the fraction: 9 27 = 3. So our expression becomes ln 3 .
Final answer Therefore, 3 ln 3 − ln 9 expressed as a single natural logarithm is ln 3 .
Examples
Logarithms are used in many scientific and engineering applications. For example, the Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. If an earthquake has a magnitude of 6 on the Richter scale, and another has a magnitude of 3, the first earthquake is 1 0 6 − 3 = 1 0 3 = 1000 times stronger. Understanding logarithm properties helps in comparing such magnitudes.
The expression 3 ln 3 − ln 9 can be simplified using logarithmic rules to give ln 3 . The chosen answer is A. ln 3 .
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