Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt{x^{2}(x+2)}\)

Short answer

The expanded form of the given logarithm expression is \(\ln x + \\frac{1}{2}\ln (x+2)\)

Step-by-step solution

Break down the square root using exponent rule

The square root of a quantity can also be expressed as raising that quantity to the power of \(\frac{1}{2}\). Therefore, the expression becomes \(\ln (x^{2}(x+2))^{\frac{1}{2}}\) which is equivalent to \(\frac{1}{2} \ln (x^{2}(x+2))\)

Use logarithmic property to break up the logs

According to the property of logarithm, logarithm of a product is the sum of the logs. Thus, we can break it down as \(\frac{1}{2}(\ln x^{2} + \ln (x+2))\)

Apply power rule

Based on the power rule, where log of \(x^{n}\) is \(n*\ln x\), we simplify the equation further into \(\frac{1}{2}(2\ln x + \ln (x+2))\)

Simplify the expression

Simplify to get the final result as \(\ln x + \frac{1}{2}\ln (x+2)\)

Key concepts

Logarithmic Expression Expansion

Understanding how to expand logarithmic expressions is essential for simplifying complex logs and solving logarithmic equations. It's much like untangling a knotted string to lay it out straight and easy to measure. Our example, \(\ln \sqrt{x^{2}(x+2)}\), is a case of such a logarithmic knot. Breaking it down involves using logarithm properties to transform a condensed log into an extended version.

To start, the property of exponentiation allows us to express the square root as an exponent of \(\frac{1}{2}\). So, our tangled log becomes \(\ln (x^{2}(x+2))^\frac{1}{2}\), which we can stretch out to \(\frac{1}{2} \ln (x^{2}(x+2))\). The action of spreading out a single logarithm into a sum, difference, or multiple of simpler logs is what we refer to as logarithmic expression expansion.

Logarithm Product Rule

Moving on to the logarithm product rule, think of it as the Swiss Army knife in simplifying logs—it splits a log of a product into a sum of logs. In our example, we apply this rule to break down the product inside the log. We have \(\ln x^{2}\) multiplied by \(\ln (x+2)\), and according to the product rule, we can write \(\frac{1}{2}\ln x^{2} + \frac{1}{2}\ln (x+2)\).

Remember, the product rule is simply put: if you have \(\ln(ab)\), you can rewrite it as \(\ln a + \ln b\). It's a helpful tool that transforms a complex problem into more manageable parts and is particularly useful in different mathematical domains such as calculus, algebra, and advanced computations involving logarithms.

Logarithm Power Rule

The logarithm power rule is yet another powerful property that helps us unravel complex logarithmic expressions. If you have \(\ln (x^n)\), this rule permits you to pull the exponent out front, rewriting the expression as \(n\ln x\). In the exercise, we apply this to \(\ln x^{2}\), which swiftly turns into \(2\ln x\) when we use the power rule.

After using the logarithm power rule, our example simplifies down to \(\ln x + \frac{1}{2}\ln (x+2)\), which looks much less intimidating than the square root we started with. The power rule is akin to decompressing a packed suitcase to see exactly what's inside, making it easier to understand and work with logarithmic functions.