Problem 18 - Algebra A Combined Function

Question

Use the power property to rewrite each expression. $$ \log _{2} x^{5} $$

Step-by-Step Solution

Answer

The expression \( \log_{2} x^{5} \) can be rewritten as \( 5 \cdot \log_{2} x \).

Step 1: Identify the Power Property of Logarithms

The power property of logarithms states that for any logarithm \( \log_b (x^n) \), we can bring down the exponent in front of the logarithm. Mathematically, it is expressed as \( \log_b (x^n) = n \cdot \log_b x \). Recognize this property for rewriting.

Step 2: Apply the Power Property

Apply the power property from Step 1 to the given expression \( \log_{2} x^{5} \). According to the property, take the exponent 5 and bring it in front of the logarithm. This gives us \( 5 \cdot \log_{2} x \).

Step 3: Rewrite the Expression

After applying the power property, rewrite the expression as \( 5 \cdot \log_{2} x \). This is the expression rewritten using the power property of logarithms.

Key Concepts

Understanding Logarithms

Logarithms are a fundamental concept in mathematics that help us work with exponential relationships more easily. Essentially, a logarithm answers the question: "To what exponent must a base be raised, to produce a given number?" If you have the equation \(b^y = x\), the logarithm form becomes \(\log_b(x) = y\). In this form, \(b\) is the base, \(x\) is the number we obtain, and \(y\) is the exponent needed to reach \(x\) from \(b\).

Logarithms are closely related to exponents. They offer a way to transform multiplication into addition, division into subtraction, and powers into multiplication, which simplifies complex calculations greatly. By understanding logs, you can handle many mathematical problems, especially those related to exponential growth or decay, such as in finance or population dynamics.

Exploring Exponents

Exponents are shorthand for repeated multiplication. For example, \(x^5\) means multiplying \(x\) by itself five times: \(x \times x \times x \times x \times x\). This compact form helps solve problems when dealing with large numbers or mathematic powers.

There are several key rules of exponents that are frequently used in calculations. For instance:

• Multiplication rule: \(x^a \times x^b = x^{a+b}\)
• Division rule: \(\frac{x^a}{x^b} = x^{a-b}\)
• Power of a power: \((x^a)^b = x^{a \cdot b}\)

Understanding how to manipulate exponents is essential in using logarithms since properties like the power property of logarithms directly relate to exponents.

Simplifying Logarithmic Expressions

Logarithmic expressions can sometimes seem complex, but using mathematical properties, they can be simplified considerably. The power property of logarithms, in particular, plays an important role in restructuring such expressions. This property states that \(\log_b(x^n) = n \cdot \log_b(x)\). Essentially, you can pull the exponent down to become a coefficient in front of the logarithm.

For example, in the expression \(\log_{2} x^{5}\), you can apply the power property to simplify it to \(5 \cdot \log_{2} x\). This transformation removes the exponent environment and makes further calculations easier. Being able to simplify logarithmic expressions makes them much more manageable in algebraic operations and problem-solving.

Key Mathematical Properties to Remember

Mathematical properties serve as tools that simplify complex expressions, including those involving logarithms. Here are some essential properties to remember:

• Product property: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
• Quotient property: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)
• Power property: \(\log_b(x^n) = n \cdot \log_b(x)\)

Each property helps rearrange expressions into a simpler form, which often makes solving equations and working with data more efficient.

By mastering these properties, students can tackle logarithmic problems with greater assurance and apply these concepts in practical scenarios such as data analysis, computer algorithms, and solving real-world problems in science and engineering. Understanding these concepts thoroughly allows for effective problem-solving and mathematical reasoning in more advanced topics.