Problem 34 - Algebra 2 and Trigonometry

Question

In \(33-44,\) if \(\ln 2=x\) and \(\ln 3=y,\) write each of the natural logs in terms of \(x\) and \(y\) $$ \ln 9 $$

Step-by-Step Solution

Answer

\( \ln 9 = 2y \), where \( y = \ln 3 \).

Step 1: Understanding the Problem

We need to express the natural logarithm \( \ln 9 \) using the given \( \ln 2 = x \) and \( \ln 3 = y \).

Step 2: Breaking Down \( \ln 9 \)

Recognize that \( 9 = 3^2 \). Therefore, we can rewrite \( \ln 9 \) as \( \ln (3^2) \).

Step 3: Using the Logarithm Power Rule

Apply the power rule of logarithms: \( \ln (a^b) = b \ln a \). Using this, \( \ln (3^2) = 2 \ln 3 \).

Step 4: Substituting Given Values

Substitute \( \ln 3 = y \) into the equation: \( 2 \ln 3 = 2y \). So, \( \ln 9 = 2y \).

Key Concepts

Logarithm Rules

Understanding logarithms starts by knowing their key rules. These rules help simplify and transform logarithmic expressions. Here are three important ones:

• Product Rule: The logarithm of a product is the sum of the logarithms: \( \log_b (MN) = \log_b M + \log_b N \).
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).
• Power Rule: Useful for powers, the logarithm of a power is the exponent times the logarithm of the base: \( \log_b (M^n) = n \cdot \log_b M \).

These rules apply to any logarithms, whether they are common logs, natural logs, or others. In the exercise, the power rule helped us express \( \ln 9 \) as \( 2 \ln 3 \). Recognizing \( 9 \) as \( 3^2 \) allowed us to use this powerful rule and break down the problem easily.

Expressing Logarithms

When expressing logarithms, the goal is to write them using known values or simpler components, as seen in the problem. Breaking down complex logarithms into terms of simpler ones is a common technique.

• Identify Simpler Components: Start by recognizing the equivalent form of the number inside the logarithm. For instance, \(9\) is the same as \(3^2\).
• Substitute Known Values: Use given logarithmic values for substitution. In the provided problem, \( \ln 2 = x \) and \( \ln 3 = y \), serving as basis for expressing \( \ln 9 \).

With these strategies, we use existing information to reformulate logarithms. This method is not only effective for simplifying expressions but also for solving logarithmic equations.

Logarithm Properties

Logarithm properties are the foundational characteristics that define how logarithms behave. These properties make it possible to manipulate and simplify complex logarithmic expressions. Here's a closer look:

• Inverse Property: Logarithms are the inverse operations of exponents. If \( b^x = N \), then \( \log_b N = x \).
• Change of Base Formula: To convert logarithmic bases, use: \( \log_b M = \frac{\log_k M}{\log_k b} \), useful when comparing logs of different bases.
• Special Values: Logarithms of 1 or same base: \( \log_b 1 = 0 \) and \( \log_b b = 1 \), provide trivially simple results.

Familiarity with these properties equips students to tackle not only textbook exercises but also more complex questions. Whether decomposing an expression or evaluating a value, logarithm properties form the core toolkit for all logarithmic manipulations.