Problem 36 - Algebra 2 and Trigonometry

Question

\(\operatorname{In} 36-47,\) if \(\log 3=x\) and \(\log 5=y,\) write each of the logs in terms of \(x\) and \(y\) $$ \log 15 $$

Step-by-Step Solution

Answer

\( \log 15 = x + y \)

Step 1: Understand the Problem

We need to express \( \log 15 \) in terms of \( x \) and \( y \), given that \( \log 3 = x \) and \( \log 5 = y \). This means we will have to relate the number 15 to 3 and 5, using the properties of logarithms.

Step 2: Express 15 in Terms of Factors 3 and 5

15 can be broken down into the product of its prime factors: \( 15 = 3 \times 5 \). We will use this factorization to express \( \log 15 \) as a combination of \( \log 3 \) and \( \log 5 \).

Step 3: Use the Product Property of Logarithms

Recall the property of logarithms: \( \log(ab) = \log a + \log b \). Applying this to \( \log(3 \times 5) \), we have \( \log(15) = \log(3) + \log(5) \).

Step 4: Substitute Given Logarithmic Values

We substitute the known values: \( \log 3 = x \) and \( \log 5 = y \) into the equation. Thus, \( \log 15 = x + y \).

Key Concepts

Product Property of Logarithms

The product property of logarithms is really helpful when you need to break down complex logarithmic expressions into simpler parts. This property states:

• \( \log(a \, b) = \log a + \log b \)

This means that the logarithm of a product is the sum of the logarithms of its components. Here’s why this is great: if you have the logarithm of a number and it's made up of smaller factors, you can represent it as a sum of simpler logarithms. Like in our exercise, where we needed \( \log 15 \) and noticed that 15 equals \( 3 \times 5 \).
By applying the product property, we can easily express this as \( \log 3 + \log 5 \). This makes solving such expressions faster and easier since it reduces complex logarithms into known values or simpler terms. Remembering this property is key to simplifying many logarithmic problems.

Logarithmic Identities

Logarithmic identities are essential tools when working with logarithms. They are rules or properties that help manipulate and simplify log expressions. The product property mentioned earlier is one such identity.Some important logarithmic identities include:

• The product property: \( \log(a \, b) = \log a + \log b \)
• The quotient property: \( \log\left(\frac{a}{b}\right) = \log a - \log b \)
• The power property: \( \log(a^b) = b \, \log a \)

Knowing these identities lets you break down and rearrange complicated logarithmic expressions into more manageable forms. For instance, in our exercise, we used the product property to express \( \log 15 \) as \( \log 3 + \log 5 \) because 15 is the product of 3 and 5. Mastering these identities strengthens your ability to handle other complex problems in mathematics, especially those involving growth and decay, sound intensity, or any multiplicative processes modeled by logs.

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors. A prime number is only divisible by 1 and itself. When you factor a number, you express it as a product of these prime elements.Here’s why prime factorization is useful in logarithms:- It simplifies the process of solving logarithmic expressions by breaking numbers into smaller, manageable parts.- It allows applying logarithmic properties effectively.In our exercise, the number 15 was decomposed into 3 and 5, since 15 = 3 \( \times \) 5. By doing so, we were able to use the product property of logarithms. This technique lets you connect the number with its simplest components, making it easier to compute or replace with known logarithmic values like \( x \) or \( y \) in this case.Prime factorization is particularly useful when dealing with logarithms because complex numbers can often be expressed as products of simpler numbers. Once you have the prime factors, using the product property becomes straightforward.