/*! 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 19 \(y=\log _{3}\left(x^{2}\right)\... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 19

\(y=\log _{3}\left(x^{2}\right)\)

Short Answer

Expert verified
The given function simplifies to \(y = 2\log_{3}(x)\).

Step by step solution

01

Understand the given function

The given function is a logarithmic function: \(y = \log_{3}(x^{2})\). This means we are finding the logarithm base 3 of the square of \(x\).
02

Apply logarithmic properties

Use the power rule of logarithms: \(\log_{a}(b^{c}) = c \log_{a}(b)\). In this case, \(a = 3\), \(b = x\), and \(c = 2\). Thus, \(y = \log_{3}(x^{2})\) becomes \(y = 2\log_{3}(x)\).

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Key Concepts

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

Logarithm Properties
Logarithm properties are essential for working with logarithmic functions. One critical property is the change of base formula, defined as \(\frac{\text{log}_a(b)}{\text{log}_a(c)} \). This helps when switching between different bases to simplify calculations.
Another important property is the addition and subtraction rules. For any positive numbers a, b, and c, with a 鈮� 1: \[ \text{log}_a{(bc)} = \text{log}_a{b} + \text{log}_a{c} \ \text{log}_a{\frac{b}{c}} = \text{log}_a{b} - \text{log}_a{c} \]
These properties allow breaking down complex expressions into simpler, more manageable parts.
Power Rule
The power rule is a logarithmic property that lets you simplify expressions involving exponents. The power rule states: \[ \text{log}_a{(b^c)} = c \text{log}_a{b} \]
This rule tells you to move the exponent in front of the logarithm. For example, given \[ \text{log}_3(x^2) \], you can transform it using the power rule to: \[ 2 \text{log}_3{x} \]
This makes the expression easier to handle and often simplifies solving equations and understanding the relationship between elements within the expression.
Base of Logarithm
In logarithmic functions, the base of the logarithm is a crucial element. It's the number you raise to a power to get another number. For instance, in \[ \text{log}_3(x^2) \], the base is 3, meaning you're figuring out the power to which 3 must be raised to yield \( x^2 \).
Changing the base can help in solving equations or graphing functions. Common bases used in logs are 10 (common logarithm) and e (natural logarithm). The natural log has base 'e', an irrational number approximately equal to 2.718. While using different bases, it鈥檚 possible to convert one base to another using the change of base formula, to standardize the calculations for easier comprehension.

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Most popular questions from this chapter

ANIMAL DEMOGRAPHY A naturalist at an animal sanctuary has determined that the function \(f(x)=\frac{4 e^{-(\ln x)^{2}}}{\sqrt{\pi} x}\) provides a good measure of the number of animals in the sanctuary that are \(x\) years old. Sketch the graph of \(f(x)\) for \(x>0\), and find the most likely age among the animals, that is, the age for which \(f(x)\) is largest.

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?

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 \(n \geq 8\) ?

\(y=x \ln x^{2}\)

RADIOACTIVE DECAY A radioactive substance decays exponentially. If 500 grams of the substance were present initially and 400 grams are prescnt 50 years later, how many grams will be present after 200 years?
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