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Chapter 5: Problem 82

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1. Assume that all variables represent positive real numbers. $$\log _{b}(2 y+5)-\frac{1}{2} \log _{b}(y+3)$$

Short Answer

Expert verified
\(\log_b\left( \frac{2y+5}{\sqrt{y+3}} \right)\).

Step by step solution

01

Identify the Logarithmic Properties

We need to use the properties of logarithms, specifically the properties involving subtraction and exponents: \( \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \) and \( a \log_b(M) = \log_b(M^a) \).
02

Apply the Power Rule

The term \(-\frac{1}{2} \log_b(y+3)\) can be rewritten using the power rule: \(-\log_b((y+3)^{\frac{1}{2}})\) or equivalently as \(\log_b\left(\frac{1}{\sqrt{y+3}}\right)\).
03

Combine the Logarithms

Use the subtraction property of logarithms to combine the expressions: \(\log_b(2y+5) + \log_b\left(\frac{1}{\sqrt{y+3}}\right) = \log_b\left( (2y+5) \times \frac{1}{\sqrt{y+3}} \right) \).
04

Simplify the Expression

Simplify the expression inside the logarithm to achieve the final form: \(\log_b\left( \frac{2y+5}{\sqrt{y+3}} \right)\). The expression is now a single logarithm with coefficient 1.

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Key Concepts

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

Logarithmic Subtraction
Logarithmic subtraction is an important concept when working with expressions involving logarithms. It helps us to simplify and condense these expressions. When you subtract logarithms with the same base, you actually perform a division within their arguments. The rule is given by:
  • \( \log_b(M) - \log_b(N) = \log_b\left( \frac{M}{N} \right) \)
Here, the term \( M\) is divided by \( N \) inside the logarithm, effectively consolidating two separate logarithmic terms into one.
For example, in the expression \( \log_b(2y+5) - \frac{1}{2} \log_b(y+3) \), this property is illustrated as part of the solution. The step involves combining the logarithms into one by utilizing this subtraction property.
Power Rule of Logarithms
The power rule of logarithms is used when there is a coefficient in front of the logarithm or when you want to work with powers within the logarithmic expression. The power rule transforms this coefficient into an exponent within the log argument. This rule is represented as follows:
  • \( a \log_b(M) = \log_b(M^a) \)
In simpler terms, the coefficient \( a \) becomes an exponent on \( M \).
For the exercise at hand, \(-\frac{1}{2} \log_b(y+3)\), the power rule helps to express it as \(-\log_b((y+3)^{\frac{1}{2}})\) or equivalently \(\log_b\left(\frac{1}{\sqrt{y+3}}\right)\).
This step is crucial as it transitions the expression into a form suitable for further application of other logarithmic properties, such as subtraction.
Simplifying Logarithmic Expressions
The goal of simplifying logarithmic expressions is to condense them into a single, more manageable form. Utilizing properties such as the power rule and subtraction allows us to rewrite complex expressions in a straightforward way.
In the given exercise, after applying the power rule and combining terms through subtraction, the expression was simplified to:
  • \( \log_b\left( (2y+5) \times \frac{1}{\sqrt{y+3}} \right) \)
This step means combining all terms into a single logarithm, ensuring the coefficient is 1.
The process involves rewriting the inside of the log function to simplify into \( \log_b\left( \frac{2y+5}{\sqrt{y+3}} \right) \). This transformation effectively reduces even complex logarithmic expressions to a neatly consolidated form. Understanding how to simplify these expressions is vital for solving equations and analyzing functions that involve logarithms.

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