Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt[3]{t}\)

Short answer

The expanded form of the expression using the properties of logarithms is \( \frac{1}{3} \ln t \).

Step-by-step solution

Apply the Power Rule of Logarithms

Following the power-rule, where \( \log_b (a^k) = k \cdot \log_b (a) \), the expression can be rewritten as: \( \ln t^{1/3} \) = \( \frac{1}{3} \ln t \).

Final Expansion

Applying step 1, the expression \( \ln \sqrt[3]{t} \) expands to \( \frac{1}{3} \ln t \). This is the final expansion of the given expression making use of the properties of logarithms.

Key concepts

Logarithm Rules

Understanding logarithm rules is crucial for expanding and simplifying logarithmic expressions. A logarithm, essentially, answers the question: to what power must the base be raised to produce a certain number? There are several important rules that govern how we manipulate logarithms.

• The Product Rule, \( \log_b(x \cdot y) = \log_b(x) + \log_b(y) \), tells us that the logarithm of a product is the sum of the logarithms.
• The Quotient Rule, \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \), shows that the logarithm of a quotient is the difference of the logarithms.
• One of the most applied properties is the Power Rule, \( \log_b (a^k) = k \cdot \log_b(a) \), which suggests that the logarithm of a power is the exponent times the logarithm of the base.
• Lastly, the Change of Base Formula, \( \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \), allows us to convert logarithms to a different base.

When solving logarithmic expressions, these rules enable us to write complex logarithms in terms of simpler ones, making it easier to expand, condense, or compute them.

For instance, the Power Rule was utilized in the given exercise to expand \(\ln \sqrt[3]{t}\) into \(\frac{1}{3} \ln t\). This is a straightforward application of the Power Rule, transforming a radical expression into a multiplication operation.

Natural Logarithm Expansion

The natural logarithm expansion is a specific case of logarithmic properties applied to the natural logarithm function, represented as \(\ln\), which has a base of Euler's number \(e\). The expansion of natural logarithms follows the same rules as those of common logarithms but is specifically applied to situations where the base is \(e\).

This form of logarithm is particularly important in disciplines such as calculus and complex mathematics, as \(e\) is a fundamental constant. For example, when expanding \(\ln \sqrt[3]{t}\) using the Power Rule, we write \(\frac{1}{3}\ln t\) instead of \(\ln (t^{1/3})\). Notice how the exponent moves out in front of the logarithm, simplifying the expression without changing its value.

Moreover, because \(e\) is an irrational number, performing expansion using natural logarithm rules enables more precise calculations when dealing with growth, decay, and continuous compounding, all of which naturally occur with base \(e\) in their mathematical models. The ability to expand and condense expressions with natural logarithms is an invaluable skill in these fields.

Logarithmic Expressions

Working with logarithmic expressions involves writing logarithms in various forms, often with the goal to simplify complex problems or to make certain features of the function more evident. Logarithms can express multiplicative patterns as additive ones and make solving exponential equations possible where the unknown is an exponent.

An understanding of how to handle logarithmic expressions is fundamental when analyzing exponential growth and decay, as well as in areas such as acoustics, information theory, and pH in chemistry, where logarithmic scales are used. Simplifying logarithms can also clarify the underlying relationships in data sets that follow a power law, and they are crucial in the study of algorithms and computing, particularly in analyzing the time complexity of algorithms.

The exercise shown earlier, which simplifies \(\ln \sqrt[3]{t}\) to \(\frac{1}{3}\ln t\), showcases a simple but important aspect of working with logarithmic expressions: using properties to reveal the direct proportionality between the logarithm and the exponent. This is not just a mathematical trick; it's a powerful tool for understanding how systems change exponentially and for communicating those changes in a clear and straightforward manner.