Problem 24 - Calculus of a Single Variable

Question

In Exercises 19–28, use the properties of logarithms to expand the logarithmic expression. $$ \ln \sqrt{a-1} $$

Step-by-Step Solution

Answer

\( 0.5 \cdot \ln (a-1) \)

Step 1: Identify Power Property of Logarithms

The power property of logarithm allows us to move an exponent to the front of the logarithm. In this case, the exponent is not immediately apparent because we have a square root sign. However, the square root of a number is equivalent to raising that number to the power of \(0.5\). So in reality, our expression looks like this: \( \ln (a-1)^{0.5} \).

Step 2: Apply Power Property of Logarithms

Applying the power property of logarithms here gives: \( 0.5\cdot \ln (a-1) \).

Step 3: Final Expression

So, the expanded form of \( \ln \sqrt{a-1} \) is given by \( 0.5 \cdot \ln (a-1) \).

Key Concepts

Power Property of Logarithms

The power property of logarithms is a fundamental rule that helps simplify logarithmic expressions by handling exponents cleanly. It states that for any positive real number \(a\), base \(b\), and any real number \(n\), the expression \(\log_b(a^n)\) is equivalent to \(n \cdot \log_b(a)\). This means you can pull the exponent out in front of the logarithm.

In the context of our problem involving natural logarithms, \(\ln\), we use this property to simplify square roots or any terms raised to a power. Recognizing that a square root is equivalent to a power of 0.5 can be incredibly useful. For instance, when faced with \(\ln \sqrt{a-1}\), rewrite it as \(\ln (a-1)^{0.5}\). Now, using the power property becomes straightforward, allowing us to move the 0.5 in front: \(0.5 \cdot \ln (a-1)\). This property can greatly simplify complex logarithmic operations by breaking them down into more manageable pieces.

Expanding Logarithmic Expressions

Expanding logarithmic expressions involves breaking down a complex logarithm into a simpler, more digestible form. This process utilizes several logarithmic rules, including the power property we discussed, as well as the product and quotient properties if applicable.

Our example focuses solely on the power property because we're transforming a square root into an exponential form. Nonetheless, whenever you expand a logarithmic expression, look for components that can be simplified or separated.

How do we expand? Consider any compound elements in the log. Identify powers, products, or quotients, and apply corresponding properties. For powers, use the power property to move the exponent to the front. If you have products (combined elements within the log), split them using \(\log_b(xy) = \log_b(x) + \log_b(y)\). For quotients, rely on \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\).

The goal is always clarity. Simplifying these components makes calculation easier and concepts clearer.

Square Roots in Logarithms

Square roots can add complexity to logarithmic expressions, but they actually offer a neat opportunity to use logarithmic properties like the power property. The trick is in understanding that a square root is a power: specifically, the power of 0.5.

When you see a square root inside a logarithm, like \(\ln \sqrt{a-1}\), recognize it as \(\ln (a-1)^{0.5}\). By expressing it this way, you pave the path for easy expansion using the power property. The exponent 0.5 can be pulled out in front of the logarithm to simplify the expression to \(0.5 \cdot \ln (a-1)\).

In pursuit of simplification, always check if the expression can be rewritten in terms of exponents. This will often make it easier to apply logarithmic rules effectively. Understanding this transformation not only aids in expanding logs but also in simplifying expressions or solving log equations.