Problem 44 - Precalculus with Limits

Question

Expanding a Logarithmic Expression In Exercises \(37-58\) , use properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\ln \sqrt[3]{t}$$

Step-by-Step Solution

Answer

The expanded logarithmic expression of \(\ln \sqrt[3]{t}\) is \((1/3)*\ln t\).

Step 1: Break down the expression

First, the immediate task is to convert the fraction exponent to represent it as a multiplier within the context of a logarithmic property. The expression \(\ln \sqrt[3]{t}\) can be written as \(\ln t^{1/3}\).

Step 2: Apply the power rule

Using the power rule in logarithms, which states that logb(m^n) = n * logb(m), we can take the fraction exponent and bring it out in front as a multiplier. This simplifies the expression to \((1/3)*\ln t\).

Step 3: Final Simplified Expression

At this point, the expression has been simplified and expanded as much as possible. There is no further breakdown or simplification possible, so \((1/3)*\ln t\) is the final simplified expression.

Key Concepts

Properties of Logarithms

Logarithms have several handy properties that make them powerful tools in simplifying and expanding expressions. These properties enable us to manipulate logarithmic expressions into easier forms. Here are the key properties you'll often use:

• Product Rule: This allows you to turn the logarithm of a product into a sum of logarithms. It states that \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• Quotient Rule: This property helps you split the logarithm of a quotient into the difference of two logs: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \).
• Power Rule: This lets you bring down powers in a logarithmic expression. For any positive number \(m\), exponent \(n\), and base \(b\), this rule states: \( \log_b(m^n) = n \cdot \log_b(m) \).

Using these properties skillfully can transform complex logarithmic expressions into more manageable parts.

Expanding Logarithms

Expanding logarithmic expressions is all about breaking them down into simpler pieces. The process often involves using properties such as the product, quotient, and power rules of logarithms.

The main goal is to express a log of a single complex expression in terms of logs of simpler pieces, typically as a series of sums, differences, or multiples. This makes calculations more straightforward and can help in solving equations or simplifying expressions further. Consider the expression \(\ln(t^{1/3})\).
By using the power rule, you can pull out the exponent as a fraction, expanding it to \(\frac{1}{3} \cdot \ln(t)\).
In many exercises, each step of expansion helps make the expression easier to work with or solve, especially when combined with other algebraic techniques like factoring.

Logarithm Power Rule

The power rule is one of the most straightforward, yet powerful, properties in logarithms. This rule says that if you have a logarithm of a number raised to a power, you can "bring down" that power in front of the log. It applies for any positive number \(m\) and any real number \(n\) in an expression like \(\log_b(m^n)\).

For example, with \(\ln(t^{1/3})\), we can use the power rule to extract \(\frac{1}{3}\) as a coefficient outside the log. This results in \(\frac{1}{3} \cdot \ln(t)\).
The usefulness of this rule lies in its ability to simplify powers inside a logarithm, making calculations and manipulations more direct and comprehensible. This is particularly helpful in calculus, algebra, and other branches of mathematics.