Write the expression as a single natural logarithm.
[tex]$3 \ln x-5 \ln c$[/tex]
A. [tex]$\ln \frac{x^3}{c^5}$[/tex]
B. [tex]$\ln \left(x^3-c^5\right)$[/tex]
C. [tex]$\ln \frac{c^5}{x^3}$[/tex]
D. [tex]$\ln \left(x^3+c^5\right)$[/tex]
Answer (2)
Apply the power rule of logarithms to rewrite the expression: 3 ln x − 5 ln c = ln x 3 − ln c 5 .
Use the quotient rule of logarithms to combine the terms into a single logarithm: ln x 3 − ln c 5 = ln c 5 x 3 .
The expression 3 ln x − 5 ln c as a single natural logarithm is ln c 5 x 3 .
Explanation
Understanding the Problem We are asked to write the expression 3 ln x − 5 ln c as a single natural logarithm. We will use logarithm properties to achieve this.
Applying the Power Rule First, we use the power rule of logarithms, which states that a ln x = ln x a . Applying this to both terms, we get: 3 ln x = ln x 3 5 ln c = ln c 5
Rewriting the Expression Now, we rewrite the original expression using these results: 3 ln x − 5 ln c = ln x 3 − ln c 5
Applying the Quotient Rule Next, we use the quotient rule of logarithms, which states that ln a − ln b = ln b a . Applying this rule, we combine the two logarithms into a single logarithm: ln x 3 − ln c 5 = ln c 5 x 3
Final Answer Therefore, the expression 3 ln x − 5 ln c can be written as a single natural logarithm: ln c 5 x 3 .
Examples
Logarithms are incredibly useful in many scientific and engineering fields. For instance, they are used to simplify calculations in acoustics, where the intensity of sound is measured on a logarithmic scale (decibels). Similarly, in chemistry, pH values, which indicate the acidity or alkalinity of a solution, are also expressed logarithmically. By understanding how to manipulate logarithmic expressions, you can easily work with and interpret data in these fields.
The expression 3 ln x − 5 ln c can be rewritten as a single natural logarithm, which is ln c 5 x 3 . This is achieved by applying the power rule and the quotient rule of logarithms. Therefore, the correct answer is option A: ln c 5 x 3 .
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