Problem 38 - 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 \(y\) $$ \ln 36 $$

Step-by-Step Solution

Answer

\(\ln 36 = 2x + 2y\)

Step 1: Express 36 as a Product of Primes

To find \(\ln 36\) in terms of \(x\) and \(y\), first express 36 as a product of its prime factors: \(36 = 2^2 \times 3^2\). This decomposition will help us break down the natural logarithm into simpler parts.

Step 2: Use Logarithm Property on \(\ln 36\)

Apply the property of logarithms that states \(\ln(a^b) = b \ln a\). So for \(\ln 36\), we can write \(\ln 36 = \ln(2^2 \times 3^2) = \ln(2^2) + \ln(3^2)\).

Step 3: Apply Power Rule to Simplify

Using the power rule of logarithms, which is \(\ln(a^b) = b\ln(a)\), simplify the expression: \(\ln(2^2) = 2\ln 2\) and \(\ln(3^2) = 2\ln 3\). Thus, \(\ln 36 = 2\ln 2 + 2\ln 3\).

Step 4: Substitute Known Values

We know from the problem statement that \(\ln 2 = x\) and \(\ln 3 = y\). Substitute these values into the equation: \(\ln 36 = 2x + 2y\).

Key Concepts

Prime Factorization

Before diving deep into logarithms, understanding prime factorization is crucial. It is the process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
For instance, when we factor 36, the aim is to express it solely with prime numbers. We divide 36 by the smallest prime number, which is 2, resulting in 18. We continue dividing by 2 until it's no longer divisible without a remainder, yielding a factor of another prime (3):

• 36 divided by 2 = 18
• 18 divided by 2 = 9
• 9 divided by 3 = 3
• 3 divided by 3 = 1

Therefore, 36 can be expressed as the product of primes: \(36 = 2^2 imes 3^2\). This prime factorization simplifies the application of logarithms in our exercise.

Logarithm Properties

Logarithms have unique properties that make complex calculations simpler. Logarithm properties are tools that manipulate and re-arrange log expressions to make calculations straightforward.
One such property is the 'product property'. It states that:

• \(\ln(a \times b) = \ln a + \ln b\)

This allows us to split the logarithm of a product into a sum of logarithms, making calculations much easier. Given our number 36 as \(2^2 \times 3^2\), we utilized this property to write:

• \(\ln(36) = \ln(2^2 \times 3^2) = \ln(2^2) + \ln(3^2)\)

Understanding this makes it easier to manage complex logarithms.

Power Rule

Another important property in the realm of logarithms is the power rule. The power rule is particularly useful when dealing with exponential expressions.
It allows us to take the exponent out of the logarithm, simplifying the expression significantly:

• \(\ln(a^b) = b \ln a\)

By using the power rule, each term in \(\ln(2^2) + \ln(3^2)\) simplifies significantly:

• For \(\ln(2^2),\) we apply the power rule to get \(2 \ln 2\).
• For \(\ln(3^2),\) the power rule yields \(2 \ln 3\).

This process turns the expression into a simpler form: \(2\ln 2 + 2\ln 3\). The power rule is a key concept for efficiently working with logarithmic equations, especially when they involve tasks like prime factorization.

Substitution Method

Finally, the substitution method comes into play when known values need to replace variables or expressions. In this exercise, substitution allows us to replace logarithmic expressions with simpler variable expressions that we already know.
Consider the problem statement which provides: \(\ln 2 = x\) and \(\ln 3 = y\). We use substitution to replace these logarithmic expressions with their respective variables:

• Instead of \(2\ln 2\), it becomes \(2x\).
• Instead of \(2\ln 3\), it becomes \(2y\).

Thus, the final expression for \(\ln 36\) in terms of \(x\) and \(y\) becomes \(2x + 2y\). Substitution is a strategy that simplifies complex algebraic expressions by replacing known values, thereby making problem-solving easier.