Problem 63 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The logarithmic expression \(\frac{1}{2}(\log x+\log y)\) can be condensed as \(\log \sqrt{xy}\)

Step 1: Apply the Logarithm Power Rule

The logarithm power rule states that \(\log_b m^n = n \log_b m\). So, we can transform the coefficient of \(\frac{1}{2}\) into a power of the logarithms: \( \frac{1}{2}(\log x + \log y) = \log x^{1/2} + \log y^{1/2}\).

Step 2: Use the Logarithm Product Rule

Next, the logarithm product rule says that \(\log_b mn = \log_b m + \log_b n\). Apply this rule in reverse to combine the two logarithms into one: \(\log x^{1/2} + \log y^{1/2} = \log (x^{1/2}y^{1/2})\) .

Step 3: Simplify the expression

Simplify the expression \(x^{1/2} y^{1/2}\) to \( \sqrt{xy} \). Therefore, \(\log (x^{1/2}y^{1/2}) = \log \sqrt{xy} \).

Key Concepts

Logarithm Power Rule

When dealing with logarithmic expressions, sometimes you'll encounter a coefficient in front of a logarithm which can be transformed using the logarithm power rule. This rule states:

• \(\log_b m^n = n \log_b m\)

This means that a coefficient isn’t just a mere decorative number – it can be incorporated into the base of the logarithm as an exponent.
In the original exercise, you have a coefficient, \( \frac{1}{2} \), in front of the expression \((\log x + \log y)\). According to the power rule, this coefficient can be moved into the logarithm's argument by applying it as a power.
When you apply this to \(\frac{1}{2}(\log x + \log y)\), it turns into \(\log x^{1/2} + \log y^{1/2}\). By doing this, we have effectively turned a multiplication outside of the logarithm into an exponent inside the logarithm.

Logarithm Product Rule

The logarithm product rule provides a straightforward way to combine two separate logarithms into one. The rule is expressed as:

• \(\log_b mn = \log_b m + \log_b n\)

By using this rule, you take advantage of the fact that adding logarithms is akin to multiplying their arguments.
In the exercise, after applying the power rule, you end up with \(\log x^{1/2} + \log y^{1/2}\). Applying the product rule in reverse, these two separate logs can be combined into one: \(\log (x^{1/2} y^{1/2})\).
This merging process simplifies the expression, turning multiple logarithms into a single concise logarithm.

Simplifying Logarithms

Once the logarithmic terms are combined and the expression is condense, the final touch to logarithmic simplification involves ensuring the result is as reduced as possible. The key here is to pay attention to mathematical simplifications that can make the logarithm more digestible.
In this specific problem, you are left with \(\log (x^{1/2}y^{1/2})\). This expression can be simplified by recognizing that \(x^{1/2}y^{1/2}\) is equivalent to \(\sqrt{x} \times \sqrt{y}\), which itself simplifies further to \(\sqrt{xy}\).
Thus, the expression \(\log (x^{1/2}y^{1/2})\) becomes \(\log \sqrt{xy}\). Simplifying in this way not only makes the logarithm easier to read, but also prepares it for further calculations or transformations, should they be required.