Problem 37 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \left[z(z-1)^{2}\right] $$

Step-by-Step Solution

Answer

The expression \( \ln \left[z(z-1)^{2}\right] \) can be written as a sum of logarithms as: \( \ln(z) + 2 \ln(z-1) \)

Step 1: Expansion using the Product Rule

Using the product rule of logarithms \( \ln(mn) = \ln(m) + \ln(n) \), the expression can be expanded as:\( \ln(z(z-1)^{2}) = \ln(z) + \ln((z-1)^{2}) \)

Step 2: Applying the Power Rule

Using the power rule of logarithms \( \ln(m^n) = n \ln(m) \), the second term in the above expression can be simplified to: \( \ln((z-1)^{2}) = 2 \ln(z-1) \). Thus, the expression becomes:\( \ln(z) + 2 \ln(z-1) \)

Key Concepts

Product Rule of Logarithms

In logarithms, the product rule is a fundamental property that simplifies the process of dealing with the logarithm of a product. Simply put, when you're working with the logarithm of a product of two numbers or expressions, you can break it down into the sum of two separate logarithms. This can be written as:\[ \ln(mn) = \ln(m) + \ln(n) \]For example, in the original exercise, we have the expression \( \ln(z(z-1)^{2}) \). Here, you can treat \( z \) and \( (z-1)^2 \) as the two separate parts of the product. Using the product rule allows us to expand this expression into:\[ \ln(z(z-1)^{2}) = \ln(z) + \ln((z-1)^{2}) \]By understanding this rule, you can simplify complex expressions more effectively. It's like splitting a complex task into smaller, more manageable parts, making it easier to work with.

Power Rule of Logarithms

Another key property in logarithms is the power rule. This rule is extremely handy when dealing with logarithms of expressions raised to a power. It states that you can bring the exponent down as a coefficient in front of the logarithm term. The rule is expressed mathematically as:\[ \ln(m^n) = n \ln(m) \]In the context of our exercise, we look at the expression \( \ln((z-1)^{2}) \). By applying the power rule, the expression can be simplified by taking the power, which is 2, and bringing it down in front of the log:\[ \ln((z-1)^{2}) = 2 \ln(z-1) \]This simplification is immensely useful because it breaks down exponential expressions within logarithms into simpler, linear terms. It often makes further calculations and manipulations straightforward.

Expansion of Logarithmic Expressions

Expanding logarithmic expressions involves using properties like the product and power rules to rewrite a single logarithmic expression as a combination of simpler logarithmic terms. This process can transform a compact form into a more understandable layout, thus easing further operations or simplifications.Consider the expression \( \ln(z(z-1)^{2}) \) again. Using the product rule, we split it into two distinct parts: \( \ln(z) + \ln((z-1)^{2}) \). Then, using the power rule, the term \( \ln((z-1)^{2}) \) is transformed further into \( 2 \ln(z-1) \). Finally, the expanded form is:\[ \ln(z) + 2 \ln(z-1) \]Through expansion, you break a potentially cumbersome expression into a series of smaller, easier-to-handle components. This method is especially beneficial when solving complex logarithmic equations, integrating logs into other mathematical functions, or preparing them for differentiation or integration.