Problem 65 - College Algebra Essentials

Question

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1)\)

Step-by-Step Solution

Answer

The single logarithmic expression is \(\log_5(\sqrt{xy} / (x + 1)^2)\).

Step 1: Apply log rule for the sum of logs

First, we apply the rule that the log of a product is the sum of the logs of its factors. This rule recognizes that \(\log_b(xy) = \log_b(x) + \log_b(y)\). Applying this rule to the first term in the given expression, we get \(\frac{1}{2}\log_5(xy)\).

Step 2: Use properties of logarithms for multiplication by a constant

Next, we apply the property that a coefficient in front of a logarithm can be rewritten as an exponent of the argument of the logarithm. That means \(\frac{1}{2}\log_5(xy)\) becomes \(\log_5((xy)^{1/2})\), or \(\log_5(\sqrt{xy})\).

Step 3: Apply the property of logs for power

Now we apply the rule \(\log_b(a^n) = n\log_b(a)\) to the second term in the expression, such that -2 \(\log_5(x+1)\) becomes \(\log_5((x+1)^{-2})\).

Step 4: Simplify expression

Lastly, we simplify the expression by combining the logs. The result is \(\log_5(\sqrt{xy} / (x + 1)^2)\). Since the division corresponds to a subtraction in the logarithms, this step is also justified by the logarithm rules.

Key Concepts

Understanding Logarithmic Expressions

Logarithmic expressions are the written form of logarithms, which are essentially the inverse operations of exponential expressions. A basic understanding of logarithms is paramount to dealing with more complex mathematical concepts. For instance, the logarithmic expression \(\log_b x\) asks the question: 'To what power must we raise the base \(b\) to receive the number \(x\)?' The importance of understanding comes into play when we need to condense or expand these expressions using various properties of logarithms.

As showcased in the original exercise, condensing a logarithmic expression involves combining multiple log terms into a single term. The process typically includes the use of several logarithm properties, but the key is to remember that each operation has a specific purpose and a strict set of rules governing it. The more comfortable you become with recognizing and applying these properties, the easier it will be to condense or expand logarithmic expressions accurately. It's like putting together a puzzle where each piece must fit perfectly based on the rules of logarithms.

Applying the Logarithm Product Rule

One crucial property that allows us to manipulate logarithms is the logarithm product rule. It states that the logarithm of a product is equal to the sum of the logarithms of each factor: \(\log_b(xy) = \log_b(x) + \log_b(y)\). In simple terms, if you have two numbers being multiplied, you can break down their logarithm into two separate terms.

When applying the logarithm product rule, as seen in the exercise, a term such as \(\frac{1}{2}(\log_5 x + \log_5 y)\) can be rewritten by combining the logs into a single log representing the square root of the product of \(x\) and \(y\), due to the coefficient \(\frac{1}{2}\). This is a transformative step that can simplify complex expressions and is often a prerequisite for solving logarithmic equations or further condensing the expressions.

Mastering the Logarithm Power Rule

The logarithm power rule is another significant tool in handling logarithms. It helps simplify expressions where a logarithm has an exponent. Formally, the power rule can be expressed as: \(\log_b(a^n) = n \log_b(a)\). This means you can take the exponent on the argument of a logarithm and ‘move’ it in front of the log.

Within the exercise context, the power rule is applied in reverse to move the coefficient in front of a log so that \( -2 \log_5(x+1) \) becomes \(\log_5((x+1)^{-2})\). This step is often used in conjunction with the product rule for further simplification. Understandably, it’s essential to gain fluency in switching between these forms through the power rule as it provides a pathway to simplify logarithmic expressions further or solve equations that would otherwise be too complicated.