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Chapter 4: Problem 20

Rewrite each expression as a sum or difference of logarithms. $$\log _{3}(x y)$$

Short Answer

Expert verified
\text{log}_{3}(x y) = \text{log}_{3}(x) + \text{log}_{3}(y)

Step by step solution

01

Identify the Logarithm Rule

Recognize that the expression \(\text{log}_{3}(xy)\) involves the product of two variables inside the logarithm. The logarithm of a product can be expressed as the sum of the logarithms of the individual factors.
02

Apply the Product Rule of Logarithms

Use the product rule of logarithms: \(\text{log}_{b}(MN) = \text{log}_{b}(M) + \text{log}_{b}(N)\). Here, identify \(M = x\) and \(N = y\).
03

Rewrite the Expression

Rewrite \(\text{log}_{3}(xy)\) using the product rule as \(\text{log}_{3}(x) + \text{log}_{3}(y)\).

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Key Concepts

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

product rule of logarithms
The product rule of logarithms is a fundamental property in logarithmic operations. This rule states that the logarithm of a product is equal to the sum of the logarithms of individual factors.
In mathematical terms, this can be written as: \[ \text{log}_{b}(MN) = \text{log}_{b}(M) + \text{log}_{b}(N) \]
Here, \(b\) represents the base of the logarithm, while \(M\) and \(N\) are the variables being multiplied inside the logarithm.
Using the product rule, we can break down complex logarithmic expressions into simpler parts, making them easier to work with.
logarithmic properties
Understanding the properties of logarithms can help you manipulate and simplify logarithmic expressions.
Some key logarithmic properties include:
  • Product Rule: As discussed earlier, \( \text{log}_{b}(MN) = \text{log}_{b}(M) + \text{log}_{b}(N) \)
  • Quotient Rule: The logarithm of a quotient is the difference of the logarithms. \[ \text{log}_{b}\frac{M}{N} = \text{log}_{b}(M) - \text{log}_{b}(N) \]
  • Power Rule: The logarithm of a power can be rewritten as the exponent times the logarithm of the base. \[ \text{log}_{b}(M^k) = k \text{log}_{b}(M) \]

By using these properties, you can transform and simplify various logarithmic expressions.
rewriting logarithms
Rewriting logarithms often involves using the fundamental properties to break down an expression or combine simpler expressions into a more complex one.
Consider the given problem: \[ \text{log}_{3}(xy) \ \] By recognizing the expression involves a product inside the logarithm, you can apply the product rule.
According to the product rule: \[ \text{log}_{3}(xy) = \text{log}_{3}(x) + \text{log}_{3}(y) \]
This process of rewriting makes it easier to handle, especially in problems requiring further simplification or when solving logarithmic equations.

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Most popular questions from this chapter

Depreciating Camry Antoinette paid \(\$ 21,075\) for a Toyota Camry. After three years it was worth \(\$ 11,981 .\) Assume that the price is decreasing according to the continuous exponential decay model \(P=P_{0} e^{r t}\). a. Find the annual depreciation rate to the nearest tenth of a percent. b. Find the value of the car after 5 years to the nearest hundred dollars.

Time of Death A detective discovered a body in a vacant lot at 7 A.M. and found that the body temperature was \(80^{\circ} \mathrm{F}\). The county coroner examined the body at 8 A.M. and found that the body temperature was \(72^{\circ} .\) Assuming that the body temperature was \(98^{\circ}\) when the person died and that the air temperature was a constant \(40^{\circ}\) all night, what was the approximate time of death?

Fill in the blank. A function of the form \(f(x)=a^{x}\) where \(a\) and \(x\) are real numbers with \(a>0\) and \(a \neq 1\) is a(n) _____ function.

Rewrite each expression in terms of \(\log _{a}(5).\) $$\log _{a}\left(5^{3}\right)$$

Richter Scale The common logarithm is used to measure the intensity of an earthquake on the Richter scale. The Richter scale rating of an earthquake of intensity \(I\) is given by \(\log (I)-\log \left(I_{0}\right),\) where \(I_{0}\) is the intensity of a small "benchmark" earthquake. Write the Richter scale rating as a single logarithm. What is the Richter scale rating of an earthquake for which \(I=1000 \cdot I_{0} ?\)
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