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Chapter 4: Problem 33

Use properties of logarithms to write each expression as a single term. $$\log x-\log (x+1)$$

Short Answer

Expert verified
\( \log \left( \frac{x}{x+1} \right) \) is the simplified single term.

Step by step solution

01

Recognize the Logarithm Rule

Identify the rule of logarithms that allows combining expressions. The rule we will use is the "quotient rule" for logarithms: \[ \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \]
02

Apply the Quotient Rule

Apply the quotient rule to combine the logarithmic expression:Given:\[ \log x - \log (x+1) \]Using the rule, you write it as:\[ \log \left( \frac{x}{x+1} \right) \]

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Key Concepts

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

Quotient Rule for Logarithms
Understanding the quotient rule is key when working with logarithmic expressions. It provides a handy shortcut for simplifying expressions where you have two logarithms being subtracted. The quotient rule for logarithms states that the difference of two logarithms with the same base is equal to the logarithm of the quotient of their arguments. Mathematically, the rule is expressed as:\[ \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \]Here's how it works:
  • Both logarithms need to have the same base.
  • The arguments of the logarithms are divided, with the argument of the first logarithm as the numerator.
  • The argument of the second logarithm becomes the denominator.
By applying the quotient rule, you can turn two separate logarithmic terms into one. This simplification helps in solving equations and simplifying expressions in a more efficient manner.
Logarithmic Expressions
To better understand logarithmic expressions, it's important to recognize them as inverse operations to exponentiation. Logarithms answer the question: "To what power must the base be raised, to produce a given number?" This can be written as:\[ b^y = x \quad \Rightarrow \quad \log_b(x) = y \]In a logarithmic expression, we have:
  • \( b \) as the base of the logarithm, typically 10 for common logarithms or \( e \) for natural logarithms.
  • \( x \) as the argument or the number we are finding the logarithm of.
  • The result of the logarithm tells us the exponent needed on the base to obtain \( x \).
Real-world applications include calculating the growth of populations in biology or the pH level in chemistry. In math problems, they often require you to manipulate logarithmic expressions, as with the problem you've tackled, to find simpler or equivalent forms.
Combining Logarithmic Terms
In mathematics, combining logarithmic terms often involves using specific properties, like the product, quotient, or power rules. This allows you to take multiple log expressions and condense them into a single logarithm, making complex equations easier to handle.With our focus here, the aim is to condense expressions by understanding:
  • When two logs are being added together, the product rule applies: \( \log_b(a) + \log_b(c) = \log_b(ac) \).
  • If subtracting, the quotient rule is used, as discussed: \( \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \).
  • The power rule is also useful: \( n \log_b(a) = \log_b(a^n) \).
Combining terms not only simplifies expressions but also helps solve logarithmic equations more effectively. Recognizing which rule to apply, like our use of the quotient rule, is critical in mathematics and problem-solving.

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