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Chapter 5: Problem 38

Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{10^{x}}{x\left(x^{2}+1\right)\left(x^{4}+2\right)}\right) $$

Short Answer

Expert verified
The expanded expression is: \(x - \log(x) - \log(x^2 + 1) - \log(x^4 + 2)\).

Step by step solution

01

Apply the Quotient Rule

The first step is to apply the quotient rule of logarithms, which states that \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \). In this expression, let \( a = 10^{x} \) and \( b = x(x^2 + 1)(x^4 + 2) \). Thus, we have:\[ \log\left(\frac{10^{x}}{x(x^2+1)(x^4+2)}\right) = \log(10^{x}) - \log\left(x(x^2 + 1)(x^4 + 2)\right). \]
02

Simplify the Logarithm of a Power

For the term \( \log(10^{x}) \), use the power rule of logarithms, which states that \( \log(a^b) = b \cdot \log(a) \). Here, \( a = 10 \) and \( b = x \). Therefore, we have:\[ \log(10^{x}) = x \cdot \log(10). \] Since \( \log(10) = 1 \), this simplifies to \( x. \)
03

Apply the Product Rule

Next, apply the product rule of logarithms to the expression \( \log\left(x(x^2 + 1)(x^4 + 2)\right) \). According to the product rule, \( \log(abc) = \log(a) + \log(b) + \log(c) \). Let \( a = x \), \( b = (x^2 + 1) \), and \( c = (x^4 + 2) \). Thus, we have:\[ \log\left(x(x^2 + 1)(x^4 + 2)\right) = \log(x) + \log(x^2 + 1) + \log(x^4 + 2). \]
04

Combine All Parts

Combine all parts to get the expanded expression. Start with \( x \), then subtract the expanded form of the denominator from Step 3:\[ \log\left(\frac{10^{x}}{x(x^2+1)(x^4+2)}\right) = x - \log(x) - \log(x^2 + 1) - \log(x^4 + 2). \]

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Key Concepts

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

Quotient Rule
The quotient rule is an essential concept in logarithms that often helps simplify complex expressions. It states that for any two positive numbers, say \( a \) and \( b \), the logarithm of their division can be written as: \[ \log\left(\frac{a}{b}\right) = \log(a) - \log(b). \]
This can be incredibly useful as it allows a division within a logarithmic function to be broken down into a subtraction operation between two separate logarithmic expressions. Let's illustrate this with an example from our exercise:
  • We start with \( a = 10^x \) and \( b = x(x^2+1)(x^4+2) \).
  • Using the quotient rule, the expression becomes: \[ \log(10^x) - \log(x(x^2 + 1)(x^4 + 2)). \]
The quotient rule makes tackling otherwise daunting logarithmic expressions much easier by converting a division into a simpler subtraction problem.
Power Rule
The power rule in logarithms is another cornerstone concept. When you have a logarithmic expression with an exponent, like \( a^b \), you can use the power rule to simplify it to: \[ \log(a^b) = b \cdot \log(a). \]
The benefit of this rule lies in how it transforms a more complex exponential form into a straightforward multiplication. For instance, in our current exercise:
  • \( a = 10 \) and \( b = x \).
  • Thanks to the power rule, \( \log(10^x) = x \cdot \log(10). \)
  • Since \( \log(10) = 1 \), the expression simplifies further to just \( x \).
This rule is particularly beneficial when dealing with exponential components as it essentially "brings down" the exponent, simplifying multiplication inside the log.
Product Rule
The product rule for logarithms helps decompose a logarithm of a product into a sum of separate logs. This rule states that for any three positive numbers \( a, b, \) and \( c \), you can write: \[ \log(abc) = \log(a) + \log(b) + \log(c). \]
This is particularly helpful when you encounter a logarithm applied to a multiplication, as it allows you to handle each part separately. In the example provided:
  • In \( \log(x(x^2+1)(x^4+2)) \), consider \( a = x \), \( b = x^2+1 \), \( c = x^4+2 \).
  • The product rule lets us break this down as: \[ \log(x) + \log(x^2 + 1) + \log(x^4 + 2). \]
Each term is now an individual logarithmic expression to work with, making complex expressions far more manageable. The product rule is handy when you need to separate intertwined multiplicative components within a logarithmic formula.

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Most popular questions from this chapter

Solve the logarithmic equation for \(x\) $$ \log _{3}(2-x)=3 $$

Find the solution of the exponential equation, correct to four decimal places. $$ 100(1.04)^{2 t}=300 $$

The population of a country has a relative growth rate of 3% per year. The government is trying to reduce the growth rate to 2%. The population in 1995 was approximately 110 million. Find the projected population for the year 2020 for the following conditions. (a) The relative growth rate remains at 3% per year. (b) The relative growth rate is reduced to 2% per year.

The fox population in a certain region has a relative growth rate of 8% per year. It is estimated that the population in 2000 was 18,000. (a) Find a function that models the population \(t\) years after 2000. (b) Use the function from part (a) to estimate the fox population in the year 2008. (c) Sketch a graph of the fox population function for the years 2000鈥�2008.

A law of physics states that the intensity of sound is inversely proportional to the square of the distance \(d\) from the source: \(I=k / d^{2} .\) (a) Use this model and the equation $$B=10 \log \frac{I}{I_{0}}$$ (described in this section) to show that the decibel levels \(B_{1}\) and \(B_{B}\) at distances \(d\) and \(d_{2}\) from a sound source are related by the equation $$B_{2}=B_{1}+20 \log \frac{d_{1}}{d_{2}}$$ (b) The intensity level at a rock concert is 120 \(\mathrm{dB}\) at a distance 2 \(\mathrm{m}\) from the speakers. Find the intensity level at a distance of 10 \(\mathrm{m} .\)
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