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Chapter 4: Problem 8

Use the properties of natural logarithms to simplify each function. $$ f(x)=\ln (4 x)-\ln 4 $$

Short Answer

Expert verified
The function simplifies to \( f(x) = \ln(x) \).

Step by step solution

01

Identify Logarithmic Rule

Recognize that we can use the logarithmic property for subtraction: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). This property allows us to combine the two logarithmic expressions into a single logarithm.
02

Apply the Logarithmic Rule

Apply the property \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \) to the given function: \( f(x) = \ln(4x) - \ln(4) \).This becomes: \( f(x) = \ln\left(\frac{4x}{4}\right) \).
03

Simplify the Expression

Simplify the expression inside the logarithm: \( \frac{4x}{4} = x \). Therefore, the function simplifies to: \( f(x) = \ln(x) \).

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Key Concepts

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

Logarithmic Properties
Logarithmic properties are fundamental rules that help simplify complex logarithmic expressions. These properties allow us to condense expressions or break them apart into simpler components.
One of the most useful properties is the subtraction rule of logarithms:
  • \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \)
  • This property simplifies the difference of two natural logarithms into a single natural logarithm.
This becomes invaluable when working with functions like the one in our original exercise where \( f(x) = \ln (4 x) - \ln 4 \). By applying this property, we can streamline the function into a simpler form.
Function Simplification
Function simplification is the process of reducing a complex mathematical function into a simpler, more manageable form. This not only makes the function easier to interpret but also simplifies further calculations.
In the context of logarithms, we use properties such as the subtraction rule to combine terms effectively.In our exercise, the function starts as \( f(x) = \ln(4x) - \ln(4) \). Using the logarithmic property:
  • Combine the terms: \( \ln\left(\frac{4x}{4}\right) \).
  • Notice how we simplify the expression inside the logarithm: \( \frac{4x}{4} = x \).
This results in the simplified function \( f(x) = \ln(x) \). Simplification helps reveal the essential behavior of the function more clearly.
Mathematical Expressions
Mathematical expressions involve various operations and symbols that describe a particular relationship. Logarithmic expressions are a type of mathematical expression that follow specific properties and rules.
When simplifying, it鈥檚 crucial to carefully manipulate the terms according to these properties.
  • The original expression \( \ln (4 x) - \ln 4 \) involves a subtraction operation of two logarithms.
  • By applying the logarithmic property, the expression is transformed, emphasizing a division operation: \( \ln\left(\frac{4x}{4}\right) \).
  • Such transformations not only simplify the expression but also decrease the risk of errors in calculations.
Through understanding how to work with these expressions, we reduce complex mathematical statements into simpler, more digestible forms, making them easier to use in further math or science applications.

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Most popular questions from this chapter

Use your graphing calculator to graph each function on the indicated interval, and give the coordinates of all relative extreme points and inflection points (rounded to two decimal places). [Hint: Use NDERIV once or twice together with ZERO.] (Answers may vary depending on the graphing window chosen.) $$ f(x)=x^{2} \ln |x| \text { for }-2 \leq x \leq 2 $$

The number of people in a city of 200,000 who have heard a weather bulletin within \(t\) hours of its first broadcast is \(N(t)=\) \(200,000\left(1-e^{-0.5 t}\right)\) a. Find \(N(0.5)\) and \(N^{\prime}(0.5)\) and interpret your answers. b. Find \(N(3)\) and \(N^{\prime}(3)\) and interpret your answers.

If consumer demand for a commodity is given by the function below (where \(p\) is the selling price in dollars), find the price that maximizes consumer expenditure. $$ D(p)=8000 e^{-0.05 p} $$

For each demand function \(D(p)\) : a. Find the elasticity of demand \(E(p)\). b. Determine whether the demand is elastic, inelastic, or unit-elastic at the given price \(p\). $$ D(p)=4000 e^{-0.01 p}, \quad p=200 $$

A $$\$ 10,000$$ automobile depreciates so that its value after \(t\) years is \(V(t)=10,000 e^{-0.35 t}\) dollars. Find the instantaneous rate of change of its value: a. when it is new \((t=0)\). b. after 2 years.
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