Problem 35 - 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 \(y\) $$ \ln 4 $$

Step-by-Step Solution

Answer

\( \ln 4 = 2x \)

Step 1: Express 4 in terms of 2

Since 4 is equal to 2 squared, we can express 4 as a power of 2: \[ 4 = 2^2 \] This will help us use the properties of logarithms to express \( \ln 4 \) in terms of \( \ln 2 \).

Step 2: Apply the Power Rule of Logarithms

The Power Rule of logarithms states that \( \ln(a^b) = b \cdot \ln a \). We can apply this rule to express the natural log of 4 as:\[ \ln 4 = \ln(2^2) = 2 \cdot \ln 2 \]

Step 3: Substitute the Known Values

We know that \( \ln 2 = x \). Substitute \( x \) into the expression:\[ \ln 4 = 2 \cdot x \] Therefore, \( \ln 4 \) is expressed as \( 2x \) in terms of \( x \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is key to simplifying complex logarithmic expressions. These properties give us the tools to rewrite logarithms in a more manageable form. One critical property is the **Product Rule**, which states that the logarithm of a product is equal to the sum of the logarithms of the factors:

• \( \ln(ab) = \ln a + \ln b \)

This lets you split logarithms of multiplied values into individual terms.
Another useful property is the **Quotient Rule**, which helps when dividing values inside a logarithm:

• \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)

Lastly, the **Power Rule** allows you to deal with exponents inside a logarithm effectively. These properties are especially useful when the intial expression is complex or when dealing with variable substitutions, allowing you to express a logarithm in different terms or variables seamlessly.

Power Rule of Logarithms

The Power Rule of logarithms is an essential concept in the realm of logarithmic properties. It is particularly useful when dealing with logarithms of values raised to a power. The rule states:

• \( \ln(a^b) = b \cdot \ln a \)

This tells us that when an argument of a logarithm is raised to an exponent, you can "bring down" the exponent and multiply the logarithm by that number.For instance, if you have \( \ln 4 \), knowing that 4 can be expressed as \( 2^2 \), the Power Rule allows you to write:

• \( \ln 4 = \ln(2^2) = 2 \cdot \ln 2 \)

This makes simplifying and expressing logarithms much easier, especially in terms of known variables or simpler expressions. Having this rule at your fingertips is crucial when solving logarithmic equations and expressing them in terms of other variables.

Expressing Logarithms in Terms of Variables

Expressing logarithms in terms of variables involves rewriting them using known values, which is a common task in math problems, like those seen in logarithmic exercises. If you know the values of certain logs involving variables, you can rewrite other logs in terms of those variables.Let's say you know \( \ln 2 = x \) and \( \ln 3 = y \). If you are given \( \ln 4 \) and need to express it in terms of \( x \), you can follow these steps:1. **Express in Known Terms**: Recognize that 4 is \( 2^2 \), which means you can apply logarithm properties to simplify it.
2. **Apply the Power Rule**: Use the Power Rule to bring down the exponent: - \( \ln(2^2) = 2 \cdot \ln 2 \) - Substitute \( x \) for \( \ln 2 \), giving you: \( 2x \)By using known logarithmic terms and properties like this, you can easily express complex expressions in terms of simpler, known variables.