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Chapter 6: Problem 84

Condense the expression to the logarithm of a single quantity. $$\ln 6+\ln y-\ln (x-3)$$

Short Answer

Expert verified
The logarithmic expression condensed to a single quantity is \( \ln\left(\frac{6y}{x-3}\right) \)

Step by step solution

01

Apply the Product Property of Logarithms

The product property of logarithms states that the logarithm of a product is the sum of the logarithms. This means that \( \ln 6 + \ln y \) can be written as \( \ln (6y) \)
02

Apply the Quotient Property of Logarithms

The quotient property of logarithms states that the logarithm of a quotient is the difference of the logarithms. This means that \( \ln (6y) - \ln (x-3) \) can be written as \( \ln\left(\frac{6y}{x-3}\right) \)
03

Write the final answer

The final condensed logarithmic expression is \( \ln\left(\frac{6y}{x-3}\right) \)

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Key Concepts

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

Product Property of Logarithms
The Product Property of Logarithms is a useful tool in simplifying logarithmic expressions. It allows us to combine two logarithms with addition into one single logarithm. Specifically, the property is expressed as: \[ \log_b (mn) = \log_b m + \log_b n \] This formula tells us that if you have the log of a product, you can break it into the sum of the logs of the factors.
  • Example: Consider \( \ln 6 + \ln y \). Using the product property, this becomes \( \ln (6y) \).
  • It applies to logs with any base, whether it鈥檚 natural log (\( \ln \)) or common log (\( \log \)).
This property is particularly helpful for condensing logarithms, which means combining several log terms into one. Remember, this only works when the logarithms are of the same base.
Quotient Property of Logarithms
The Quotient Property of Logarithms is another important concept for working with logarithms. This property states that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. It is represented by the equation: \[ \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \] Let鈥檚 break it down with an example:
  • Suppose we have \( \ln (6y) - \ln (x-3) \). By using the quotient property, this can be rewritten as \( \ln \left(\frac{6y}{x-3}\right) \).
  • This is helpful when you need to condense log expressions or when performing simplifications in logarithmic equations.
It鈥檚 important to verify that both logs have the same base before applying this property. This property significantly simplifies expressions, especially in solving algebraic problems involving logarithms.
Condensing Logarithmic Expressions
Condensing logarithmic expressions is all about simplifying a series of log terms into a more manageable form鈥攁 single logarithm. This process often makes complex expressions easier to analyze and solve. Here鈥檚 how to do it using log properties:
  • Combine products: Use the product property to convert sums of logs into a log of a product.
  • Combine quotients: Apply the quotient property to turn differences of logs into a log of a quotient.
Imagine you have the expression \( \ln 6 + \ln y - \ln (x-3) \).
  • First, using the product property, \( \ln 6 + \ln y \) becomes \( \ln (6y) \).
  • Then, with the quotient property, combine this with \( \ln (x-3) \) to get \( \ln \left(\frac{6y}{x-3}\right) \).
The final result is a single, condensed logarithmic expression. This not only simplifies calculations but also helps in understanding the underlying relationships between components in mathematical problems.

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Most popular questions from this chapter

\(r=5+4 \cos \theta\)

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Hyperbola } & (4, \pi / 2),(1, \pi / 2) \end{array} $$

\(r=2 \csc \theta+5\)

In Exercises 73-78, solve the trigonometric equation. $$ 9 \csc ^{2} x-10=2 $$

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Parabola } & (1,-\pi / 2) \\ \end{array} $$
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