/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 \(y=\ln \sqrt{x^{2}+4 x+1}\)... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 18

\(y=\ln \sqrt{x^{2}+4 x+1}\)

Short Answer

Expert verified
The simplified form of the function is \( y = \frac{1}{2} \ln(x^2 + 4x + 1) \).

Step by step solution

01

Identify the Inner Function

Examine the function inside the natural logarithm. Here, the inner function is \( \sqrt{x^2 + 4x + 1} \).
02

Simplify the Inner Function

Rewrite the square root as an exponent: \( \sqrt{x^2 + 4x + 1} = (x^2 + 4x + 1)^{1/2} \).
03

Apply the Natural Logarithm Properties

Use the property of the natural logarithm: \( \ln(a^b) = b \ln(a) \). Therefore, \( \ln((x^2 + 4x + 1)^{1/2}) = \frac{1}{2} \ln(x^2 + 4x + 1) \).
04

Final Expression

Combine the steps to write the final simplified expression: \( y = \frac{1}{2} \ln(x^2 + 4x + 1) \).

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Key Concepts

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

Logarithmic Properties
Understanding logarithmic properties is crucial for simplifying expressions like natural logarithms. One fundamental property is the power rule: \( \ln(a^b) = b \ln(a) \). \This property allows us to handle logarithms more flexibly by moving the exponent outside the logarithm.\It simplifies complex expressions, which makes them easier to differentiate or integrate later.\Remember to always reduce the expression inside the logarithm as much as possible before applying this rule. \This helps in simplifying the problem further.
Inner Function
When simplifying natural logarithms, it's important to identify and understand the inner function. In the given exercise, the inner function is \( \sqrt{x^2 + 4x + 1} \).\Identifying the inner function helps in breaking down the original problem into more manageable parts. \Here, we first recognize the term \( x^2 + 4x + 1 \) under the square root.\Then, by rewriting the square root as \( (x^2 + 4x + 1)^{1/2} \), we simplify the form and prepare it for logarithmic properties application.
Algebraic Manipulation
Algebraic manipulation involves rewriting expressions to make them simpler. To complete the given exercise, we change \(\root{x^2 + 4x + 1}\) into \( (x^2 + 4x + 1)^{1/2} \). \This step is crucial because it converts the root into an exponent, enabling the application of logarithmic properties.\It may seem minor, but this step forms the foundation for applying further mathematical concepts effectively. \Always inspect your expressions to check if they can be rewritten in a more simplified or useful form.
Natural Logarithm
The natural logarithm, \( \ln(x) \), has some unique properties that can simplify mathematical problems. As shown in the exercise, understanding properties like \( \ln(a^b) = b \ln(a) \) makes complex expressions easier to manage.\In this case, after simplifying the inner function accurately, you can straightaway utilize this logarithmic property.\The final expression \( y = \frac{1}{2} \ln(x^2 + 4x + 1) \) showcases how natural logarithm properties are applied to simplify the equation.\These logarithmic manipulations are incredibly useful in algebra and calculus to solve and simplify equations.

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Most popular questions from this chapter

Make a table for the quantities \((\sqrt{n})^{\sqrt{n+1}}\) and \((\sqrt{n+1})^{\sqrt{n}}\), with \(n=8,9,12,20,25,31,37\), \(38,43,50,100\), and 1,000 . Which of the two quantities seems to be larger? Do you think this inequality holds for all \(n \geq 8\) ?

\(y=\frac{e^{-2 x}\left(2-x^{3}\right)^{3 / 2}}{\sqrt{1+x^{2}}}\)

BACTERIAL GROWTH The number of bacteria in a certain culture grows exponentially. If 5,000 bacteria were initially present and 8,000 were present 10 minutes later, how long will it take for the number of bacteria to double?

GROWTH OF BACTERIA The following data were compiled by a researcher during the first 10 minutes of an experiment designed to study the growth of bacteria: \begin{tabular}{l|c|c} Number of minutes & 0 & 10 \\ \hline Number of bacteria & 5,000 & 8,000 \end{tabular} Assuming that the number of bacteria grows exponentially, how many bacteria will be present after 30 minutes?

POPULATION GROWTH It is estimated that \(t\) years from now the population of a certain country will be \(P\) million people, where $$ P(t)=\frac{30}{1+2 e^{-0.05 t}} $$ a. Sketch the graph of \(P(t)\). b. What is the current population? c. What will be the population in 20 years? d. What happens to the population in the long run?
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