Problem 13 - Precalculus

Question

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log \left(\frac{100 x \sqrt{y}}{\sqrt[3]{10}}\right) $$

Step-by-Step Solution

Answer

The expanded and simplified form is: \(2 + \log(x) + \frac{1}{2} \log(y) - \frac{1}{3}\).

Step 1: Apply the Quotient Rule for Logarithms

The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Apply this rule: \[\log \left(\frac{100x \sqrt{y}}{\sqrt[3]{10}}\right) = \log (100x \sqrt{y}) - \log (\sqrt[3]{10})\]

Step 2: Expand the Logarithm of the Numerator Product

Use the product rule for logarithms, where the logarithm of a product is the sum of the logarithms:\[\log (100x \sqrt{y}) = \log (100) + \log (x) + \log (\sqrt{y})\]

Step 3: Simplify Each Logarithmic Expression

Simplify the logs individually:- \(\log(100) = 2\) since \(100 = 10^2\).- \(\log(\sqrt{y}) = \log(y^{1/2}) = \frac{1}{2} \log(y)\).- \(\log(\sqrt[3]{10}) = \log(10^{1/3}) = \frac{1}{3} \log(10) = \frac{1}{3}\) since \(\log(10) = 1\).Collectively, we have:\[2 + \log(x) + \frac{1}{2} \log(y) - \frac{1}{3}\]

Step 4: Write the Final Expanded and Simplified Form

Combine all the simplified components into the final expression:\[\log \left(\frac{100x \sqrt{y}}{\sqrt[3]{10}}\right) = 2 + \log(x) + \frac{1}{2} \log(y) - \frac{1}{3}\]

Key Concepts

Quotient Rule for Logarithms

The quotient rule for logarithms is a key tool when expanding expressions that involve division inside a logarithm. This rule states that the logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator. This is expressed in the formula: \[ \log\left(\frac{A}{B}\right) = \log(A) - \log(B) \]Let's take a closer look at its application using an example. If you have an expression like \(\log \left(\frac{100x \sqrt{y}}{\sqrt[3]{10}}\right)\), the first step is to separate the expression using the quotient rule. This gives \( \log(100x \sqrt{y}) - \log(\sqrt[3]{10}) \),clearly breaking down the original expression into two simpler parts that can be further expanded or simplified.

• This rule is helpful because it transforms a complex division problem into a simpler subtraction problem.
• Remember to apply this rule when you encounter a division inside a log function, as it allows each part to be individually addressed.

Product Rule for Logarithms

Once you've applied the quotient rule, another important tool is the product rule for logarithms. This rule simplifies an expression inside a logarithm when you have a product. The product rule is represented as:\[ \log(AB) = \log(A) + \log(B) \]In the context of our example, the expression \(\log(100x \sqrt{y})\) can be expanded using the product rule. This becomes: \( \log(100) + \log(x) + \log(\sqrt{y}) \).

• The benefit of this rule is that it splits a multiplication inside a logarithm into separate logarithms that are added together.
• It allows each factor of the product to be calculated and simplified individually, making the task much easier.

By using the product rule, larger expressions are broken into smaller, manageable pieces, making the entire simplification process straightforward.

Simplifying Logarithmic Expressions

After applying both the quotient and product rules, the goal is to simplify each component to make the expression as clean as possible. Let's simplify each part of our example:

• For \(\log(100)\), since \(100 = 10^2\), it simplifies to \(2\). This is because \(\log(10^2) = 2\times \log(10) = 2\), with \(\log(10) = 1\).
• For \(\log(\sqrt{y})\), which is \(\log(y^{1/2})\), it simplifies to \(\frac{1}{2} \log(y)\).
• For \(\log(\sqrt[3]{10})\), this is \(\log(10^{1/3})\), which results in \(\frac{1}{3}\) since \(\log(10) = 1\).

The final simplified expression for our initial logarithmic expression becomes:\[ 2 + \log(x) + \frac{1}{2} \log(y) - \frac{1}{3} \]This process of simplification makes complex logarithmic expressions more manageable and expresses them in their simplest form. Simplification is critical, as it condenses a potentially complicated series of terms into an easier-to-handle expression. Every simplified function represents a clearer insight into the problem, aiding in further mathematical operations or analysis.