Problem 17 - Algebra A Combined Function

Question

Use the power property to rewrite each expression. $$ \log _{3} x^{2} $$

Step-by-Step Solution

Answer

\( 2 \log_{3} x \)

Step 1: Identify the Power Property of Logarithms

The power property of logarithms states that \( \log_b (x^a) = a \log_b (x) \). This property allows us to move the exponent to the front as a multiplier of the log.

Step 2: Apply the Power Property

Apply the power property to the expression \( \log_{3} x^{2} \). According to the property, it can be rewritten as \( 2 \log_{3} x \).

Step 3: Final Expression

The expression \( \log_{3} x^2 \) has been rewritten using the power property as \( 2 \log_{3} x \).

Key Concepts

Logarithms

Logarithms are mathematical tools that help us solve equations involving exponential growth or decay. At their core, a logarithm identifies the power to which a base number must be raised to obtain another number. For example, in base 10, the logarithm of 100 is 2, because 10 squared is 100. Understanding logarithms allows us to manipulate and solve complex problems involving powers, as they can transform multiplicative relationships into additive ones. This transformation is particularly useful in algebra and calculus.

There are several key properties of logarithms to be aware of:

• Product Property: This states that the logarithm of a product is the sum of the logs of its factors. Mathematically, \( \log_b (xy) = \log_b (x) + \log_b (y) \).
• Quotient Property: The log of a quotient is the difference of the logs. If you have \( \log_b \left( \frac{x}{y} \right) = \log_b (x) - \log_b (y) \).
• Power Property: This allows us to bring the exponent down as a multiplier. As stated, \( \log_b (x^a) = a \cdot \log_b (x) \).

Understanding these properties makes solving logarithmic equations much simpler.

Exponents

Exponents are another powerful mathematical concept that deal with repeated multiplication. When you see something like \(x^2\), it means "multiply \(x\) by itself". Exponents are invaluable for expressing very large or very small numbers succinctly. They are integral to algebra, calculus, and beyond.

Here are some essential properties of exponents:

• Product of Powers Property: When multiplying two expressions with the same base, you add the exponents: \(x^a \times x^b = x^{a+b}\).
• Power of a Power Property: When raising an exponent to another exponent, you multiply the exponents: \( (x^a)^b = x^{ab} \).
• Power of a Product Property: Distributing the exponent over a product involves raising each factor to the power: \( (xy)^a = x^a \cdot y^a \).

Understanding the behavior of exponents allows us to harness the power of algorithms and solve exponential equations effectively.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. The symbols represent numbers and quantities in formulae and equations. At its most basic level, algebra involves working with unknown variables to solve equations, which can be anything from simple linear equations to more complex quadratic and polynomial equations.

Key strategies when working with algebra include:

• Simplification: Combine like terms and use distributive properties to make equations easier to tackle.
• Equation Solving: Use inverse operations to isolate variables and solve equations.
• Factoring: Break down expressions into factors to simplify multiplication and division.

Algebra is essential for higher-order math courses. By learning it, you gain the tools needed to understand more complex topics like calculus, statistics, and beyond.