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Chapter 3: Problem 79

In Exercises \(67-82,\) condense the expression to the logarithm of a single quantity. $$\frac{1}{3}\left[2 \ln (x+3)+\ln x-\ln \left(x^{2}-1\right)\right]$$

Short Answer

Expert verified
\(ln\left( \left( \frac{(x+3)^2 * x}{x^2 - 1}\right)^\frac{1}{3}\right)\)

Step by step solution

01

Apply Power Rule

Apply the power rule to the expression inside the brackets. This means the 2 coefficient in front of \(ln(x+3)\) can be placed inside the \(ln\) as a power to give:\n\n1/3[ \(ln((x+3)^2) + ln(x) - ln(x^2 - 1)\) ]
02

Apply Product and Quotient Rule

Recognize that the expression now has three logarithms that can be combined into one logarithm using the properties of logarithms. The addition will turn into multiplication and the subtraction will turn into division to give: \n\n1/3[\(ln\left( \frac{(x+3)^2 * x}{x^2 - 1}\right)\) ]
03

Expand Expression and Simplify

Simplify the expression by applying the 1/3 as a power to the whole expression. Result: \n\n\(ln\left( \left( \frac{(x+3)^2 * x}{x^2 - 1}\right)^\frac{1}{3}\right)\)

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Key Concepts

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

Power Rule in Logarithms
The Power Rule in logarithms is a handy tool for simplifying expressions. It tells us that if you have a number multiplying a logarithm, you can move that number to become an exponent on the term inside the logarithm.
This rule is expressed as \( b \, \log_a(M) = \log_a(M^b) \).
  • In our example, the coefficient 2 in front of \( \ln(x+3) \) is moved inside to become \( \ln((x+3)^2) \).
This not only simplifies the expression but also makes it easier to combine with other logarithms later on.
Remember, the Power Rule is useful in condensing multiple terms into a single logarithm, allowing for a more straightforward understanding of complex logarithmic expressions.
Product Rule in Logarithms
The Product Rule in logarithms helps you to combine terms when you have the sum of two logarithms.
This rule states that if you have two logs with the same base being added, you can multiply the arguments inside the logs.
Mathematically, it can be written as \( \log_a(M) + \log_a(N) = \log_a(M \cdot N) \).
  • In our situation, after applying the Power Rule, we have \( \ln((x+3)^2) + \ln(x) \).
  • According to the Product Rule, this becomes \( \ln((x+3)^2 \cdot x) \).
This rule allows us to condense the addition of logarithms into a single term, simplifying the expression and making further manipulation easier.
Quotient Rule in Logarithms
The Quotient Rule in logarithms is a useful technique when dealing with the subtraction of two logarithms. It allows us to rewrite the expression as a single logarithm by dividing the values inside the logs.
The rule is given by \( \log_a(M) - \log_a(N) = \log_a\left( \frac{M}{N} \right) \).
  • Applying this to our exercise, especially after dealing with the Product Rule, we have \( \ln((x+3)^2 \cdot x) - \ln(x^2 - 1) \).
  • Using the Quotient Rule, this expression can now be condensed to \( \ln\left( \frac{(x+3)^2 \cdot x}{x^2 - 1} \right) \).
Using the Quotient Rule simplifies complex expressions by reducing the number of logarithms, making it much easier to evaluate or further manipulate.

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Most popular questions from this chapter

Home Mortgage \(A \$ 120,000\) home mortgage for 30 years at \(7 \frac{1}{2} \%\) has a monthly payment of \(\$ 839.06\) Part of the monthly payment covers the interest charge on the unpaid balance, and the remainder of the payment reduces the principal. The amount paid toward the interest is $$u=M-\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}$$ and the amount paid toward the reduction of the principal is $$v=\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}$$ In these formulas, \(P\) is the size of the mortgage, \(r\) is the interest rate, \(M\) is the monthly payment, and \(t\) is the time (in years). (a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 30 years of mortgage payments.) (b) In the early years of the mortgage, is the greater part of the monthly payment paid toward the interest or the principal? Approximate the time when the monthly payment is evenly divided between interest and principal reduction. (c) Repeat parts (a) and (b) for a repayment period of 20 years \((M=\$ 966.71) .\) What can you conclude?

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