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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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Most popular questions from this chapter

Write each equation in exponential form. $$\log _{e}(54.598) \approx 4$$

The radioactive element sodium-24 has a half-life of 15 hr and is used to help locate obstructions in blood flow. If the procedure requires \(0.75 \mathrm{g}\) and is scheduled to take place in 2 days \((48 \mathrm{hr}),\) what minimum amount must be on hand now (to the nearest hundredth of a \(\operatorname{gram})^{2}\)

Solve each equation using the uniqueness property of logarithms. $$\ln (x-1)+\ln 6=\ln (3 x)$$

Intensity of sound: The intensity of sound as perceived by the human ear is measured in units called decibels (dB). The loudest sounds that can be withstood without damage to the eardrum are in the 120 - to 130 -dB range, while a whisper may measure in the 15 - to 20 -dB range. Decibel measure is given by the equation \(D(I)=10 \log \left(\frac{I}{I_{0}}\right),\) where \(I\) is the actual intensity of the sound and \(I_{0}\) is the faintest sound perceptible by the human earcalled the reference intensity. The intensity \(I\) is often given as a multiple of this reference intensity, but often the constant \(10^{-16}\) (watts per cm \(^{2}\); \(\left.\mathrm{W} / \mathrm{cm}^{2}\right)\) is used as the threshold of audibility. Sound intensity of a hair dryer: Every morning (it seems), Jose is awakened by the mind-jarring, ear-jamming sound of his daughter's hair dryer \((75 \mathrm{dB}) .\) He knew he was exaggerating, but told her (many times) of how it reminded him of his railroad days, when the air compressor for the pneumatic tools was running \((110 \mathrm{dB}) .\) In fact, how many times more intense was the sound of the air compressor compared to the sound of the hair dryer?

Solve each equation. Write answers in exact form and in approximate form to four decimal places. $$\frac{1}{2} \ln (2 x+5)+3=3.2$$
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