Problem 41 - 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 \(y\) $$ \log 27 $$

Step-by-Step Solution

Answer

\( \log 27 = 3x \)

Step 1: Understanding the Given Expression

We need to express \( \log 27 \) in terms of \( x \) and \( y \), where \( \log 3 = x \) and \( \log 5 = y \). First, observe that 27 can be expressed as a power of 3. Specifically, \( 27 = 3^3 \).

Step 2: Applying Logarithm Power Rule

Use the power rule of logarithms which states that \( \log a^b = b \log a \). Applying this to \( \log 27 = \log(3^3) \), we get \( \log 27 = 3 \log 3 \).

Step 3: Substitute the Known Value

Now substitute the given value \( \log 3 = x \) into the expression \( \log 27 = 3 \log 3 \). This gives us \( \log 27 = 3x \).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical expressions that involve logarithms. A logarithm is essentially the opposite operation of exponentiation. If you understand exponents, you have a grasp on the basics needed to understand logarithms. An expression like \(\log 27\) asks us to find the exponent to which the base must be raised in order to achieve a certain number. In this case, it’s asking to which power we need to raise 10 (or any base commonly used, but typically 10 for common logs) to get 27.

• In any logarithmic expression, you have the base and the result.
• For example, \(\log_b n\) asks, "To what power must we raise \(b\) to get \(n\)?"

Recognizing these parts and knowing how each operates can make working with logarithms much easier.
To convert or express logarithm in terms of other variables or known logs, it's crucial to start by recognizing the equivalent exponential expressions.

Power Rule of Logarithms

The power rule of logarithms is a useful property when dealing with expressions involving exponents inside a logarithm. This rule states that \(\log(a^b) = b \log(a)\). Understanding this rule simplifies many complex logarithmic expressions.
For the exercise provided, we need to express \(\log 27\) in terms of \(x\) and \(y\), where \(\log 3 = x\).
Since 27 can be rewritten as \(3^3\), the power rule allows us to transform \(\log 27\) into \(\log(3^3)\).

• Applying the power rule gives \(3 \log 3\).
• Since \(\log 3 = x\), substitute \(x\) for \(\log 3\), resulting in \(3x\).

This shows how the power rule of logarithms can quickly help simplify what might otherwise be a more complicated computation.

Expressing Logarithms

Expressing logarithms explicitly in terms of known variables allows for easier calculations, especially when solving more complicated mathematics problems. For instance, in our example problem, expressing \(\log 27\) required simplification and substitution.

• First, recognize how to express the number inside the log using prime factorization or known base relations.
• Use known logarithmic properties and rules to simplify the expression, as done with the power rule.
• Substitute any given values to convert the logarithm into terms of the known variables.

By understanding and applying these steps, you can express a wide variety of complex logarithmic expressions in simpler terms, enabling you to solve more complex problems efficiently. In practice, expressing logarithms in terms of \(x\) and \(y\) simplifies numerous mathematical operations and calculations.