Problem 36 - 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 \(y\) $$ \ln 12 $$

Step-by-Step Solution

Answer

\(\ln 12 = 2x + y\)

Step 1: Factorize the number 12

First, express 12 as a product of its prime factors. We have 12 = 2² × 3. This will help us rewrite the natural logarithm in terms of x and y.

Step 2: Use the properties of logarithms

Using the property of logarithms that states \( \ln(a \times b) = \ln a + \ln b \), we can say:\[\ln 12 = \ln(2^2 \times 3) = \ln(2^2) + \ln(3).\]

Step 3: Apply the power rule of logarithms

The power rule of logarithms states that \( \ln(a^b) = b \cdot \ln a \). Apply this rule to \( \ln(2^2) \) to get:\[\ln(2^2) = 2 \cdot \ln 2.\]

Step 4: Substitute the known values

Now substitute \( \ln 2 = x \) and \( \ln 3 = y \) into the equation obtained from Step 2 and Step 3. Thus, we have:\[\ln 12 = 2x + y.\]

Step 5: Write the final expression

The expression for \( \ln 12 \) in terms of \( x \) and \( y \) is given by:\[\ln 12 = 2x + y.\]

Key Concepts

Properties of Logarithms

Logarithms have some really interesting properties that make them incredibly useful in simplifying mathematical expressions.
One of these properties is the product rule.
This rule states that the logarithm of a product is equal to the sum of the logarithms of its individual factors.

• Mathematically, this can be presented as: \[\ln(a \times b) = \ln a + \ln b\]
• This property is particularly helpful when you break down complex numbers into smaller ones, making calculations easier.

In our exercise, we used this property by expressing \(\ln 12\) as \(\ln(2^2 \times 3)\), which simplifies to \(\ln(2^2) + \ln(3)\).
Understanding this property is your key to simplifying the computation of logarithmic expressions that may not initially seem straightforward.

Prime Factorization

Before using logarithmic properties, it's essential to understand the concept of prime factorization.
Prime factorization is the process of expressing a number as the product of its prime factors.

• This means that a number like 12 can be broken down as \(2^2 \times 3\) because 2 and 3 are prime numbers.
• These prime numbers are the building blocks of all numbers.

By using prime factorization, you can work with smaller, more manageable numbers in logarithmic expressions.
The exercise showed this by first factorizing 12 as \(2^2 \times 3\), allowing us to use properties of logarithms efficiently.
Memorizing basic prime factors or practicing more complex factorizations can save you a lot of time when simplifying logarithmic expressions.

Power Rule of Logarithms

The power rule of logarithms is another useful tool.
This rule states that the logarithm of a number raised to a power is the power times the logarithm of the number.

• The formula is: \[\ln(a^b) = b \cdot \ln a\]
• Such a rule makes dealing with exponents straightforward by turning them into multipliers.

In the exercise, we applied the power rule to \(\ln(2^2)\).
Applying the rule, we find that \(\ln(2^2) = 2 \cdot \ln 2\), simplifying the original problem significantly.
This approach allows for smoother computation and understanding of complex logarithmic expressions, especially in solving equations or inequalities involving logs.