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Chapter 5: Problem 69

Condense the expression to the logarithm of a single quantity. \(\ln x-3 \ln (x+1)\)

Short Answer

Expert verified
The expression condenses to \( \ln\frac {x}{(x+1)^3} \)

Step by step solution

01

Apply Coefficient as Exponent

Begin by leveraging the logarithmic property that allows us to rewrite the coefficient as an exponent within the logarithm: \(\ln x - 3 \ln (x+1) = \ln x - \ln (x+1)^3 \)
02

Combine Logs via Subtraction-Division Rule

Next, utilize the logarithmic property that permits redefining a subtraction of logarithms as division within a single log: \(\ln x - \ln (x+1)^3 = \ln \frac {x}{(x+1)^3} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Properties
Understanding logarithmic expressions is crucial as they offer a way to manipulate and simplify exponential problems.
When working with logs, certain properties help in transforming and condensing expressions.
Here are a few key properties:
  • Product Rule: \(\log_b(MN) = \log_b M + \log_b N\)
  • Quotient Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\)
  • Power Rule: \(\log_b(M^p) = p\cdot\log_b M\)
In the exercise we reviewed, we particularly dealt with the Power Rule, allowing us to rewrite a term by moving the coefficient up as an exponent within the logarithm.
Coefficient as Exponent
One of the most useful properties in logarithms is changing a coefficient into an exponent. This is known as the "Power Rule."
Whenever you have a term like \(a \log_b M\), you can rewrite it as \(\log_b(M^a)\).For the given example, \3 \ln (x+1)\ was transformed into \(\ln ((x+1)^3)\).
This makes manipulation easier when you want to combine terms, especially in subtraction or division.
Subtraction-Division Rule
When handling logs, subtraction has a special relation to division through the "Quotient Rule."
This rule states that when you subtract two logs with the same base, \(\log_b A - \log_b B\), you can combine them into \(\log_b \left(\frac{A}{B}\right)\).In the solution we explored, \(\ln x - \ln (x+1)^3\) was combined using this rule, resulting in \(\ln \frac{x}{(x+1)^3}\).
This not only simplifies expressions but also provides a concise form useful in calculus and other mathematical applications.

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Most popular questions from this chapter

\(g(x)=\log _{a} x \quad x=a^{2}\)

\(f(x)=\ln (3-x)\)

\(\ln (x+2)=\ln 6\)

Students in a mathematics class were given an exam and then retested monthly with an equivalent exam. The average scores for the class are given by the human memory model \(f(t)=80-17 \log (t+1)\), \(0 \leq t \leq 12\) where \(t\) is the time in months. (a) Use a graphing utility to graph the model over the specified domain. (b) What was the average score on the original exam \((t=0) ?\) (c) What was the average score after 4 months? (d) What was the average score after 10 months?

\(e^{1 / 3}=1.3956 \ldots\)
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