Problem 39 - Algebra 2 and Trigonometry

Question

Write each expression in terms of \(A\) and \(B\) if \(\log _{2} x=A\) and \(\log _{2} y=B\). \(\log _{2} x y^{3}\)

Step-by-Step Solution

Answer

The expression is written as \( A + 3B \).

Step 1: Apply the Logarithm Product Rule

The expression involves a product of two terms, \(x\) and \(y^3\). According to the logarithm product rule, \( \log_b (mn) = \log_b m + \log_b n \). Using this rule, we can rewrite \(\log_{2}(xy^3)\) as \(\log_{2} x + \log_{2} y^3\).

Step 2: Apply the Power Rule to the Second Term

The second term, \(\log_{2} y^3\), is a power expression. By the power rule for logarithms, \(\log_b (m^n) = n \cdot \log_b m \). Thus, \(\log_{2} y^3\) becomes \(3 \cdot \log_{2} y\).

Step 3: Substitute Given Logarithms

We now substitute the given values for \(\log_{2} x\) and \(\log_{2} y\). From the problem statement, we know \(\log_{2} x = A\) and \(\log_{2} y = B\). Substitute these into the expression: \(A + 3B\).

Step 4: Combine the Results

Combine the results from the previous steps to write the original expression in terms of \(A\) and \(B\). We have \(\log_{2}(xy^3) = A + 3B\).

Key Concepts

Understanding the Logarithm Product Rule

The Logarithm Product Rule is an essential concept in logarithmic expressions. It's used when multiplying two numbers inside a logarithm. This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors. Formally, it can be expressed as:

• \( \log_b (mn) = \log_b m + \log_b n \)

Here, \( b \) is the base of the logarithm, while \( m \) and \( n \) are the numbers being multiplied. This rule helps simplify expressions by breaking down complex logarithms into easier-to-handle parts.

Applying this to our initial expression \( \log_2(x y^3) \) allows us to express it as:

• \( \log_2 x + \log_2 y^3 \)

This separation makes it much easier to work with the expression, especially when combined with other rules like the power rule.

Exploring the Logarithm Power Rule

The Logarithm Power Rule comes into play whenever you see a logarithm with an exponent. It's a powerful tool for simplifying expressions where the argument of the logarithm is raised to a power. This rule is expressed as:

• \( \log_b (m^n) = n \cdot \log_b m \)

This means that you can "bring the exponent down" and multiply it by the logarithm.

For the expression \( \log_2 y^3 \), the exponent 3 is brought in front of the logarithm, transforming it to:

• \( 3 \cdot \log_2 y \)

This transformation significantly simplifies the process of expressing equations, making the calculations straightforward and manageable. It's especially useful when dealing with expressions involving powers and helps in reducing the potential for error.

The Change of Base Formula Demystified

The Change of Base Formula is an essential tool for solving logarithms with bases other than those found on standard calculators, typically base 10 or base \( e \). It allows you to convert a logarithm of any base \( b \) into a simple division of logarithms with a different base. This formula can be expressed as:

• \( \log_b a = \frac{\log_k a}{\log_k b} \)

where \( k \) is the new base. This formula is particularly useful because it provides flexibility and allows you to solve logarithms in terms of a base that is more convenient for calculation, often ensuring compatibility with technological tools.

While the task in our exercise does not require using the Change of Base Formula, understanding it prepares you for more advanced problems, enabling seamless calculation across different base systems.