Problem 31 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\log 6.3-\log 3$$

Step-by-Step Solution

Answer

\(\log(2.1)\)

Step 1: Applying the properties of logarithms

The given expression can be rewritten using the properties of logarithms. Specifically, we can use the property that states \(\log a - \log b = \log (\frac{a}{b})\). So, we can rewrite the expression as follows: \(\log 6.3 - \log 3 = \log (\frac{6.3}{3})\).

Step 2: Simplifying the Result

To simplify the expression \(\log (\frac{6.3}{3})\), we divide \(6.3\) by \(3\) to get \(2.1\). So, the simplified form of the original expression is \(\log(2.1)\).

Key Concepts

Understanding the Properties of Logarithms

Logarithms have unique properties that help in manipulating and simplifying expressions. One of the most important properties is the quotient rule, given by \( \log a - \log b = \log \left( \frac{a}{b} \right) \). This rule allows us to combine two logarithms into one by dividing the contents of the logs.

Consider the expression \( \log 6.3 - \log 3 \). Using the quotient rule, we can rewrite it as a single log: \( \log \left( \frac{6.3}{3} \right) \).

This property is crucial because it helps us simplify complex logarithmic expressions into more manageable forms.

• Quotient Rule: Used for differences of logs.
• Product Rule: Used for sums; \( \log a + \log b = \log (a \times b) \).
• Power Rule: Applies to exponents; \( \log (a^b) = b \cdot \log a \).

Exploring Logarithmic Expressions

Logarithmic expressions involve the logarithm function, which is the inverse of exponentiation. They are often used in equations to solve for unknown exponents.

When dealing with logarithmic expressions like \( \log 6.3 - \log 3 \), it's helpful to recognize that these can be simplified using their properties.

In this exercise, the objective is to express multiple logarithms as a single, simplified logarithm. By rewriting as a single logarithmic expression, we make the calculation easier and cleaner.

• Focus on combining logs using properties.
• Simplification reduces calculation errors.
• Ensures expressions are ready for further mathematical manipulation.

Simplifying Logarithms Effectively

Simplifying logarithms is a vital skill that enhances understanding and computation efficiency.

After rewriting the expression \( \log \left( \frac{6.3}{3} \right) \) using the quotient rule, simplify further by calculating the fraction inside the log:
\[ \frac{6.3}{3} = 2.1 \]
Thus, \( \log (2.1) \) is the simplified form of the original expression.

This step ensures that your final result is as concise as possible.

• Perform arithmetic inside the log first.
• Focus on simplifying complex expressions.
• Make sure the expression is defined for all variable values.