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Chapter 4: Problem 5

Use the properties of natural logarithms to simplify each function. $$ f(x)=\ln (9 x)-\ln 9 $$

Short Answer

Expert verified
The simplified function is \( f(x) = \ln x \).

Step by step solution

01

Understanding the Expression

We start with the given expression: \( f(x) = \ln (9x) - \ln 9 \). Our goal is to use properties of natural logarithms to simplify this.
02

Applying Logarithmic Property

We use the property of logarithms that states \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \). In our expression, \( a = 9x \) and \( b = 9 \).
03

Simplifying the Expression

Applying the property, we get: \( f(x) = \ln \left( \frac{9x}{9} \right) \).
04

Final Simplification

Simplifying further, the expression becomes \( f(x) = \ln x \) since \( \frac{9x}{9} = x \).

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Key Concepts

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

Logarithmic Properties
Natural logarithms, often denoted as \( \ln \), are widely used in mathematics, particularly in calculus and algebra. One of the key properties of logarithms is the ability to simplify expressions involving division.
  • For two positive numbers \( a \) and \( b \), the property \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \) allows us to condense the difference between two logs into a single logarithmic expression.
  • This means that when a function like \( \ln(9x) - \ln 9 \) is present, it can be rewritten as \( \ln \left( \frac{9x}{9} \right) \).
By using this property, expressions can be greatly simplified, transforming complex-looking logarithmic expressions into more manageable forms.
Simplification of Expressions
Simplifying expressions is a fundamental part of working with logarithms. It often involves reducing expressions to their simplest terms.
In the case of \( f(x) = \ln(9x) - \ln 9 \), we see how understanding the rules of logarithms can simplify the expression.
  • First, apply the property we discussed, \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \), to get \( \ln \left( \frac{9x}{9} \right) \).
  • Next, simplify the fraction: \( \frac{9x}{9} \) simplifies to \( x \), giving us the simplified form \( \ln x \).
These steps show how complex expressions can often be broken down with the right properties, resulting in simpler, more elegant functions.
Calculus Problem Solving
In calculus, logarithmic functions and their properties are crucial tools for solving various problems.
The simplification of logarithmic expressions is not just a neat algebraic trick; it plays a significant role in calculus, especially when dealing with derivatives and integrals.
  • Simplified forms like \( \ln x \) are easier to work with when performing calculus operations, such as taking a derivative or an integral.
  • For instance, the derivative of \( \ln x \) is \( \frac{1}{x} \), a simple expression that arises directly from the property-based simplification we did before.
By simplifying logarithmic expressions, we set ourselves up for much clearer and straightforward calculus operations. Understanding these principles enhances your problem-solving skills and efficiency when tackling complex calculus problems.

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Most popular questions from this chapter

Derive the formula $$ \log _{a} x=\frac{\ln x}{\ln a} \quad(\text { for } a>0 \text { and } x>0) $$ which expresses logarithms to \(a n y\) base \(a\) in terms of natural logarithms, as follows: a. Define \(y=\log _{a} x\), so that \(x=a^{y}\), and take the natural logarithms of both sides of the last equation and obtain \(\ln x=y \ln a\). b. Solve the last equation for \(y\) to obtain \(y=\frac{\ln x}{\ln a}\) and then use the original definition of \(y\) to obtain the stated change of base formula.

Explain why it is obvious, without any calculation, that \(\frac{d}{d x} \ln e^{x}=1\).

A recent study found that one's earnings are affected by the mathematics courses one has taken. In particular, compared to someone making $$\$ 40,000$$ who had taken no calculus, a comparable person who had taken \(x\) years of calculus would be earning \(\quad\) $$\$40,000 e^{0.195 x}$$. Find the rate of change of this function at \(x=1\) and interpret your answer.

A European oil-producing country estimates that the demand for its oil (in millions of barrels per day) is \(D(p)=3.5 e^{-0.06 p}\), where \(p\) is the price of a barrel of oil. To raise its revenues, should it raise or lower its price from its current level of $$\$ 120$$ per barrel?

For each function: a. Find the relative rate of change. b. Evaluate the relative rate of change at the given value(s) of \(t\). $$ f(t)=e^{t^{3}}, \quad t=5 $$
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