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Chapter 4: Problem 22

Use the Laws of Logarithms to expand the expression. $$\log _{5} \frac{x}{2}$$

Short Answer

Expert verified
The expanded form is \( \log_{5}(x) - \log_{5}(2) \).

Step by step solution

01

Understand the logarithm function

The expression given is a logarithm of a fraction, \( \log_{5}\left(\frac{x}{2}\right) \). The task is to expand it using the laws of logarithms. In this case, we will use the law that deals with division within a logarithmic expression.
02

Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \). Applying this to \( \log_{5}\left(\frac{x}{2}\right) \), we get \( \log_{5}(x) - \log_{5}(2) \).

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Key Concepts

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

Laws of Logarithms
Logarithms operate under a set of powerful rules that simplify complex expressions, known as the "laws of logarithms." These laws allow us to manipulate logarithmic expressions, making them easier to solve or interpret. The three main laws include:
  • Product Rule: This rule states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, it's expressed as \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \).
  • Quotient Rule: This involves division and states that the logarithm of a quotient is the difference between the logarithm of the numerator and the denominator. It's written as \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \).
  • Power Rule: This rule handles exponents inside a log and simplifies them by bringing the exponent out as a multiplier: \( \log_b(M^k) = k \cdot \log_b(M) \).
These laws are essential in transforming logarithmic expressions into more manageable forms, especially when solving equations or expanding expressions.
Quotient Rule
The quotient rule is specifically used when dealing with the division of numbers inside a logarithm. In the given problem, \( \log_{5}\left( \frac{x}{2} \right) \), this rule becomes very handy. This rule transforms the division under the log sign into a subtraction operation of two logs.When using the quotient rule, think of the expression as essentially "splitting" the division into subtraction, reducing the complexity:
  • The original expression is \( \log_{5}\left( x \right) \) minus \( \log_{5}\left( 2\right) \).
This rule is particularly beneficial when you're trying to break down a log expression into its individual components or when isolating variables in equations.
Logarithmic Expansion
Logarithmic expansion involves breaking down a complex logarithmic expression into simpler terms, using the laws of logarithms. For our exercise, the goal was to expand \( \log_{5}\left( \frac{x}{2} \right) \).Here's how it goes:
  • Start with identifying which log law applies: In this case, the Quotient Rule is ideal for expanding \( \log_{5}\left( \frac{x}{2} \right) \).
  • Apply this rule to get \( \log_{5}(x) - \log_{5}(2) \). Now, the expression is simplified into two separate logs, making it easier to handle and understand.
Expansion like this is crucial in algebra and calculus because it allows for the separation of variables and individual term investigation. It forms a foundation for more advanced manipulations and solutions of logarithmic equations.

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Most popular questions from this chapter

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If \(D_{0}\) is the original amount of carbon-14 and \(D\) is the amount remaining, then the artifact's age \(A\) (in years) is given by $$A=-8267 \ln \left(\frac{D}{D_{0}}\right)$$ Find the age of an object if the amount \(D\) of carbon- 14 that remains in the object is \(73 \%\) of the original amount \(D_{0}\).

Compare the rates of growth of the functions \(f(x)=\ln x\) and \(g(x)=\sqrt{x}\) by drawing their graphs on a common screen using the viewing rectangle \([-1,30]\) by \([-1,6]\).

Doubling an Investment How long will it take for an investment of \(\$ 1000\) to double in value if the interest rate is \(8.5 \%\) per year, compounded continuously?

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The magnitude \(M\) of a star is a measure of how bright a star appears to the human eye. It is defined by $$M=-2.5 \log \left(\frac{B}{B_{0}}\right)$$ where \(B\) is the actual brightness of the star and \(B_{0}\) is a constant. (a) Expand the right-hand side of the equation. (b) Use part (a) to show that the brighter a star, the less its magnitude. (c) Betelgeuse is about 100 times brighter than Albiero. Use part (a) to show that Betelgeuse is 5 magnitudes less bright than Albiero.
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