Problem 39 - 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 \(y\) $$ \ln \frac{1}{3} $$

Step-by-Step Solution

Answer

\(\ln \left(\frac{1}{3}\right) = -y\)

Step 1: Recall Logarithmic Properties

One of the properties of logarithms is \(\ln \left(\frac{1}{a}\right) = -\ln a\). This will help us express the given logarithm \(\ln \left(\frac{1}{3}\right)\) in a simpler form.

Step 2: Apply the Property

Using the property from Step 1, substitute \(a=3\) into the formula. We get:\[ \ln \left(\frac{1}{3}\right) = -\ln 3 \]

Step 3: Substitute Given Values

From the problem statement, we know \(\ln 3 = y\). Substitute \(y\) for \(\ln 3\) in the equation from Step 2:\[ \ln \left(\frac{1}{3}\right) = -y \]

Key Concepts

Logarithmic Properties

Logarithmic properties are rules that apply to logarithmic expressions, making them easier to manipulate. These properties are essential for simplifying complex logarithm problems. Here are some important logarithmic properties that you should know:

• The Product Rule: For any positive numbers \(a\) and \(b\), the logarithm of a product is the sum of the logarithms. Mathematically, it's expressed as \(\log_b(ab) = \log_b(a) + \log_b(b)\).
• The Quotient Rule: For any positive numbers \(a\) and \(b\), the logarithm of a quotient is the difference of the logarithms. It can be expressed as \(\log_b\left(\frac{a}{b}\right) = \log_b(a) - \log_b(b)\).
• The Power Rule: For any positive number \(a\) and any real number \(c\), the logarithm of a power is equal to the power times the logarithm. This means \(\log_b(a^c) = c \cdot \log_b(a)\).

In the **original exercise**, the property used was the logarithm of a reciprocal: \(\ln\left(\frac{1}{a}\right) = -\ln a\). This rule is particularly useful when you need to convert division problems into subtraction problems, simplifying the logarithmic expressions.

Natural Logarithms

Natural logarithms are a special type of logarithm where the base is the mathematical constant \(e\) (approximately 2.718). Natural logarithms are denoted by \(\ln\), short for 'logarithmus naturalis,' the Latin for natural logarithm. They are commonly used in higher mathematics and natural sciences, as \(e\) represents continuous growth or decay, found in many natural phenomena.

Here are some key points about natural logarithms:

• Natural logarithms follow the same properties as any other logarithms, such as the product, quotient, and power rules.
  
  
• In contexts involving exponential growth and decay, such as population growth or radioactive decay, natural logarithms often appear as they provide a natural way to describe these processes.
• For example, when you see \(\ln(2) = x\), it means that \(\ln\) is being used to express the exponent needed to get 2 when using the base \(e\).

Understanding **natural logarithms** helps in interpreting processes subdued to exponential growth, making them ubiquitous in biology, chemistry, and economics.

Logarithm Conversion

Logarithmic conversion is the process of changing from one logarithmic form to another, often to make equations easier to solve. This is especially useful when switching between different bases.

For any two positive numbers \(a\) and \(b\) with the same base \(b\), you can use the change of base formula to convert between logarithms of different bases:

• \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\)

This formula helps you convert between logarithms that appear in different contexts.

In the **original exercise**, \(\ln \left(\frac{1}{3}\right)\) was converted using the logarithmic properties to express it in terms of \(x\) and \(y\). By recognizing the given \(\ln 3 = y\), we could convert \(\ln \left(\frac{1}{3}\right)\) into \(-y\) efficiently.

• Knowing how to convert between different logarithmic forms is crucial, as calculations involving \(ln\), \(log_2\), or \(log_{10}\) often require translating between these expressions.

Being comfortable with **logarithm conversion** techniques can streamline your math problems and broaden your ability to interpret logarithmic information in various fields like engineering and physics.