Problem 37 - 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 \(y\) $$ \log 9 $$

Step-by-Step Solution

Answer

\(\log 9 = 2x\)

Step 1: Understanding the Expression

The goal is to express \( \log 9 \) in terms of \( x \) and \( y \), where \( \log 3 = x \). We recognize that 9 is a power of 3, specifically 9 = 3^2.

Step 2: Using Logarithm Properties

Apply the power rule of logarithms, which states \( \log(a^b) = b \cdot \log a \). Here, \( a = 3 \) and \( b = 2 \), so \( \log 9 = \log(3^2) = 2 \cdot \log 3 \).

Step 3: Substituting Known Values

Since \( \log 3 = x \), substitute \( x \) into the expression: \( 2 \cdot \log 3 = 2 \cdot x \).

Step 4: Final Expression

Thus, \( \log 9 \) can be expressed in terms of \( x \) as \( 2x \).

Key Concepts

Logarithm Properties

Logarithms are a fascinating concept in mathematics that allow us to solve equations involving exponents. When working with logarithms, there are several key properties that can aid in simplifying and rewriting expressions. These properties include:

• **Product Property**: \( \log(b \cdot c) = \log b + \log c \) – This property allows you to split the logarithm of a product into the sum of individual logarithms.
• **Quotient Property**: \( \log\left(\frac{b}{c}\right) = \log b - \log c \) – This property helps break down the logarithm of a quotient into a difference.
• **Power Property**: \( \log(a^b) = b \cdot \log a \) – This is particularly useful for dealing with powers inside a logarithm.

Each of these properties helps to simplify complex logarithmic expressions into more manageable forms. Understanding these rules is essential for manipulating expressions and for translating logarithmic statements into something more usable. These tools are particularly helpful when you want to express a logarithm in terms of known values, such as variables like \( x \) and \( y \).
Knowing when and how to apply these properties is a crucial skill in algebra and calculus, allowing you to solve a broad range of problems.

Power Rule of Logarithms

The power rule of logarithms is a particularly handy tool when dealing with expressions where the logarithm encompasses an exponent. The power rule states that \( \log(a^b) = b \cdot \log a \), which effectively "pulls out" the exponent as a coefficient.For instance, if we have \( \log(3^2) \), applying the power rule gives us \( 2 \cdot \log 3 \). This simplifies our work significantly, as we need only deal with the logarithm of the base number, rather than the exponentiated value.

The power rule is especially critical in exercises where you need to express a logarithm in terms of a given variable. In our exercise, 9 was rewritten using exponents: \( 9 = 3^2 \). Using the power rule allowed us to express \( \log 9 \) in terms of \( \log 3 \), which we already knew equaled \( x \). Thus, employing the power rule led seamlessly to the expression \( \log 9 = 2 \cdot x \).
Mastering the power rule allows for efficient problem-solving and is foundational for manipulating more advanced algebraic expressions.

Expressing Logarithms in Terms of Variables

Expressing logarithms in terms of variables is a common technique in algebra that allows us to simplify expressions and solve equations. This method involves taking the known values of specific logarithms (expressed as variables) and using them to transform other logarithmic expressions.In our example, we started with the information that \( \log 3 = x \) and explored how to express \( \log 9 \) in terms of \( x \). This process required recognizing that 9 can be rewritten as a power of 3 (i.e., \( 9 = 3^2 \)), allowing us to apply the power rule of logarithms. With \( \log 9 = 2 \cdot \log 3 \), substituting \( \log 3 \) with \( x \) gives us \( \log 9 = 2x \).

This technique can be extensively used by mathematizing real-world problems into solvable forms using logarithmic properties. By expressing complex logarithmic functions in terms of simpler, known variables, solving equations becomes significantly more manageable. It's a technique that not only simplifies calculations but also opens the door to understanding relationships between different logarithmic expressions.