Exhaustive Study Guide: Properties and Expansion of Logarithms

The Quotient Property of Natural Logarithms: When expanding a natural log that involves division, such as $\ln\left(\frac{x}{y}\right)$ , the expression is separated using subtraction. * The property follows the structure: $\ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y)$ . * The numerator must always come first in the expanded expression; order is critically important in this operation.
Understanding Logarithmic Bases: * Natural Log (ln): The base of $\ln$ is understood to be $e$ . * Common Log (log): The base of a regular $\log$ is understood to be $10$ if no base is explicitly written. * In both cases, if the base is not specified, it is conventional not to write it.

General Rule: A core principle of expanding logarithms is that no logarithm should retain an exponent in its final expanded form.
The Property Statement: $\log_{a}(u^{n}) = n \times \log_{a}(u)$ .
The Procedure: If a term within a logarithm has an exponent, that exponent is moved to the front of the logarithm as a multiplier.
Integration with Other Rules: When expanding complex expressions, the general steps suggest separating terms by addition or subtraction first, then moving exponents as the final step.

Problem: Expand $\log(x^{3} y^{2})^{4}$ .
Step 1: Simplify Exponents: Apply the "power to a power" rule. Multiply the outer exponent by the internal exponents. * $3 \times 4 = 12$ * $2 \times 4 = 8$ * The expression becomes $\log(x^{12} y^{8})$ .
Step 2: Separate by Operation: Since $x^{12}$ and $y^{8}$ are multiplied, separate them using addition. * $\log(x^{12}) + \log(y^{8})$ .
Step 3: Move Exponents: Move the exponents to the front as the final step. * Final Answer: $12 \log(x) + 8 \log(y)$ .

Problem: Expand $\ln\left(\frac{x}{y^{3}}\right)^{4}$ .
Step 1: Distribute the Exponent: Bring the power of $4$ to both the numerator and the denominator. Note that $x$ has an implicit power of $1$ . * $1 \times 4 = 4$ * $3 \times 4 = 12$ * The expression becomes $\ln\left(\frac{x^{4}}{y^{12}}\right)$ .
Step 2: Separate by Operation: Use subtraction to separate the division. * $\ln(x^{4}) - \ln(y^{12})$ .
Step 3: Move Exponents: Finalize the expansion by moving the exponents to the front. * Final Answer: $4 \ln(x) - 12 \ln(y)$ .

Problem: Expand $\log_{4}\left(\frac{x^{3}}{y z^{2}}\right)$ .
Step 1: Identify Operations: The numerator contains $x^{3}$ . The denominator contains a product: $y \times z^{2}$ .
Step 2: Separate with Grouping: Separate the numerator and denominator using subtraction, but group the denominator terms together because they are being multiplied. * $\log_{4}(x^{3}) - (\log_{4}(y) + \log_{4}(z^{2}))$
Step 3: Distribute the Negative: Distribute the negative sign from the subtraction into the parentheses. * $\log_{4}(x^{3}) - \log_{4}(y) - \log_{4}(z^{2})$ .
Step 4: Bring Down Exponents: Perform the final step of moving exponents to the front. * Final Answer: $3 \log_{4}(x) - \log_{4}(y) - 2 \log_{4}(z)$ .

Problem: Expand $\log_{6}\left(\frac{\sqrt{x} y^{3}}{z^{5}}\right)$ .
Conversion of Radicals: Radicals are not permitted in the final expanded answer. They must be rewritten as fractional exponents using the "power over root" rule. * Square root of $x$ is written as $\sqrt{x} = x^{1/2}$ . * Cube root (hypothetically) would be written as $x^{1/3}$ .
Step 1: Separate All Terms: Treat the numerator products with addition and the denominator with subtraction. * $\log_{6}(\sqrt{x}) + \log_{6}(y^{3}) - \log_{6}(z^{5})$ .
Step 2: Rewrite Radical as Exponent: * $\log_{6}(x^{1/2}) + \log_{6}(y^{3}) - \log_{6}(z^{5})$ .
Step 3: Final Power Move: Bring all exponents to the front. * Final Answer: $\frac{1}{2} \log_{6}(x) + 3 \log_{6}(y) - 5 \log_{6}(z)$ .

Student Question: Does the power rule apply only to that specific property?
Instructor Response: Yes, it applies only if there is an exponent. If there is no multiplication or division, you don't use those properties, but if an exponent is present during expansion, it must move to the front.
Discussion on Difficulty: The instructor describes logarithms as "puzzle pieces" that simply shift around, suggesting the process is not inherently difficult but requires moving pieces correctly.
Logistic Exchange: * Student: "Can I go [to the bathroom] after?" * Instructor: "Mhmm. Alright. Don't be overwhelmed. All of them are on the same page."
Future Lesson Plans: The instructor mentions that instead of a worksheet today, students will practice these specific examples. Tomorrow's notes will be of similar length (four examples), covering the same logic of shifting parts of the logarithmic expressions.