Rewriting Log₁₅2³ Using The Power Property Of Logarithms

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In the realm of mathematics, logarithms serve as a fundamental tool for simplifying complex calculations and revealing relationships between numbers. Among the various properties of logarithms, the power property stands out as a particularly useful technique for manipulating logarithmic expressions. This article delves into the application of the power property to rewrite the logarithmic expression log₁₅2³, providing a comprehensive explanation and step-by-step guidance.

The power property of logarithms states that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. Mathematically, this can be expressed as:

logₐ(xⁿ) = n * logₐ(x)

Where:

  • a is the base of the logarithm (a > 0 and a ≠ 1)
  • x is a positive real number (x > 0)
  • n is any real number

This property allows us to bring exponents outside the logarithm, which can simplify expressions and make them easier to work with.

In this case, we are tasked with rewriting the expression log₁₅2³ using the power property. Here's how we can do it:

  1. Identify the base, the number, and the power.

    • The base of the logarithm is 15.
    • The number is 2.
    • The power is 3.

  2. Apply the power property.

    Using the power property, we can rewrite the expression as:

    log₁₅2³ = 3 * log₁₅2

Let's break down the application of the power property step by step to ensure clarity:

  1. Original Expression: We start with the expression log₁₅2³.
  2. Identify the Components: We identify the base as 15, the number as 2, and the exponent as 3.
  3. Apply the Power Property: We bring the exponent 3 down and multiply it by the logarithm of 2 with base 15. This gives us 3 * log₁₅2.

Thus, log₁₅2³ rewritten using the power property is 3 * log₁₅2.

The power property is a valuable tool in various mathematical contexts, particularly in simplifying expressions and solving equations involving logarithms. Here are some reasons why it's useful:

  • Simplification: It allows us to simplify complex logarithmic expressions by reducing exponents, making them easier to work with.
  • Equation Solving: It helps in solving equations where the variable is part of an exponent within a logarithm.
  • Calculations: It simplifies calculations by breaking down complex expressions into simpler terms.
  • Applications: It is crucial in fields like exponential growth and decay, where logarithmic scales are used to measure large quantities.

Consider the equation:

log₂(x⁴) = 8

To solve for x, we first use the power property to rewrite the equation:

4 * log₂(x) = 8

Next, divide both sides by 4:

log₂(x) = 2

Then, convert the logarithmic equation to an exponential equation:

x = 2²

x = 4

This example clearly demonstrates how the power property aids in simplifying and solving logarithmic equations.

When applying the power property, it's crucial to avoid common mistakes that can lead to incorrect results. Here are a few to keep in mind:

  • Misunderstanding the Property: The power property applies only when the entire argument of the logarithm is raised to a power, not just part of it. For example, log₂(x³) is not the same as (log₂x)³.
  • Incorrectly Applying the Property: Ensure that you multiply the logarithm by the power, not add or subtract it.
  • Forgetting the Base: Always keep the base of the logarithm in mind, as it affects the final result.
  • Confusing with Other Properties: Do not confuse the power property with other logarithmic properties such as the product or quotient rule.

To solidify your understanding of the power property, try these practice problems:

  1. Rewrite log₃(9⁵) using the power property.
  2. Simplify log₄(16²).
  3. Solve for x: log₅(x³) = 6.

By working through these problems, you'll gain confidence in applying the power property and manipulating logarithmic expressions.

The power property of logarithms is a fundamental tool for simplifying and manipulating logarithmic expressions. By understanding and applying this property, you can rewrite expressions like log₁₅2³ in a more manageable form. In this specific case, log₁₅2³ is rewritten as 3 * log₁₅2. Mastering the power property is essential for anyone working with logarithms, as it simplifies complex calculations and aids in solving a variety of mathematical problems. This knowledge not only enhances your problem-solving skills but also provides a deeper understanding of the mathematical relationships inherent in logarithmic functions. Remember to practice and apply this property in various contexts to fully appreciate its power and utility. The ability to manipulate logarithmic expressions effectively opens doors to more advanced mathematical concepts and applications in various scientific and engineering fields.

The correct answer is D. 3 log₁₅2.