Problem 19
Question
\(y=\log _{3}\left(x^{2}\right)\)
Step-by-Step Solution
Verified Answer
The given function simplifies to \(y = 2\log_{3}(x)\).
1Step 1: 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 \(x\).
2Step 2: 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 \(y = 2\log_{3}(x)\).
Key Concepts
Logarithm PropertiesPower RuleBase of Logarithm
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 ≠ 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.
Another important property is the addition and subtraction rules. For any positive numbers a, b, and c, with a ≠ 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.
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’s possible to convert one base to another using the change of base formula, to standardize the calculations for easier comprehension.
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’s possible to convert one base to another using the change of base formula, to standardize the calculations for easier comprehension.