Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{6} z^{-3}$$.

Step-by-Step Solution

Answer

The expanded form of \(\log_6 z^{-3}\) is \(-3 \log_6 z\)

Step 1: Applying the power law

Using the power law of logarithms, \(\log_b a^n = n \log_b a\), where a and b are positive real numbers and n is a real number, the given expression \(\log_6 z^{-3}\) can be rewritten as: \(-3 \log_6 z\)

Step 2: Final result

The logarithmic expression has been expanded as a constant multiple of a logarithm. The final result is \(-3 \log_6 z\)

Key Concepts

Logarithmic Expressions

When dealing with logarithmic expressions, it is important to understand the basics of logarithms. A logarithm represents the power to which a number, called the base, must be raised to obtain another number. For example, in the expression \( \log_b a \), \(b\) is the base, and \(a\) is the number we want to find the power for. The result of a logarithmic function is the exponent that makes the base \(b\) equal to \(a\).

Understanding how to manipulate these expressions is crucial for solving many mathematical problems. They appear frequently in various fields such as science, engineering, and economics. Therefore, being able to skillfully convert logarithmic expressions into their expanded or condensed forms can be highly beneficial. It's like having a conversational language with numbers, where being fluent allows you to express complex numerical ideas in simpler terms.

Expanding Logarithms

The process of expanding logarithms is similar to translating a dense paragraph into a list with bullet points. Each component of the logarithmic expression is broken down into individual parts. This often comes in handy to simplify complex logarithmic equations into more manageable pieces. To expand logarithms, we use properties that define how these mathematical expressions operate.

For instance, the product property \( \log_b(mn) = \log_b m + \log_b n \) tells us that a logarithm of a product can be expressed as the sum of two logarithms. Similarly, the quotient property \( \log_b(m/n) = \log_b m - \log_b n \) allows us to turn the logarithm of a division into the subtraction of two logarithms. Understanding these properties helps in transforming a condensed expression into an expanded one, making it clearer and often more straightforward to work with.

Power Law of Logarithms

The power law of logarithms is crucial when working with logarithmic expressions, especially when the input to the log function is raised to an exponent. This law states that \(\log_b a^n = n \log_b a\), where \(n\) is the exponent. This means that if you have a logarithm of a number that is raised to a power, like \(z^{-3}\) in the example, you can 'pull out' the exponent and make it a multiplier for the entire log expression.

Using this law makes the logarithmic expression more flexible and easier to integrate into algebraic operations. It's a bit like unpacking a suitcase to see clearly what's inside. In the given exercise, by utilizing the power law, we simplified \(\log_6 z^{-3}\) to become \( -3 \log_6 z \), unravelling the exponent and bringing it to the forefront. The exponent no longer hides within the confines of a logarithmic function, making subsequent steps in the solution clearer and often much simpler.