Problem 41 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log 5+\log 2 $$

Step-by-Step Solution

Answer

\(\log 5+\log 2 = \log 10\).

Step 1: Understanding Property of Logarithms

In order to condense the expression, it is essential to understand the properties of logarithms. Here, the Product Rule of Logarithms comes into play, which states that \(\log_a b + \log_a c = \log_a (b×c)\). This property is true for any real numbers \(b\) and \(c\), and for any log base \(a\).

Step 2: Apply the Property of Logarithms

Applying the property to the given expression \(\log 5+\log 2\), it can be written as a single logarithm of the product \(5×2\). So we write it as, \(\log 5+\log 2 = \log (5*2)\).

Step 3: Evaluate the Expression

Now, calculate the product inside the logarithm to get the final expression. Compute \(5×2\) to get \(10\). Hence, \(\log (5*2) = \log 10\).

Key Concepts

Properties of Logarithms

Logarithms are a powerful tool in mathematics, especially when dealing with large numbers, multiplication, and division. One of the key aspects to understand about logarithms is their properties. These properties help to simplify and manipulate expressions involving logarithms. Some of the crucial properties include:

• Product Rule: When you add two logarithms with the same base, it is equivalent to the logarithm of the product of those two numbers. Mathematically, this is expressed as \( \log_a b + \log_a c = \log_a (b \times c) \).
• Quotient Rule: The subtraction of two logarithms with the same base can be rewritten as the log of the division of the two terms. This is \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \).
• Power Rule: Multiplying a logarithm by a number can be moved as a power on the inside of the logarithm, shown as \( n \cdot \log_a b = \log_a (b^n) \).

These properties may initially seem abstract, but they are incredibly useful for simplifying complex logarithmic expressions. This lesson focuses on the Product Rule, which directly applies to our exercise and solution.

Product Rule

The Product Rule is one of the most intuitive logarithmic properties for dealing with additions in logarithmic expressions. Let's explore how the Product Rule works. This rule states that the logarithm of a product is the sum of the logarithms of the individual factors. Therefore, the expression \( \log_a b + \log_a c \) can be simplified to a single logarithmic term: \( \log_a (b \times c) \). In our example, the task was to condense \( \log 5 + \log 2 \). Using the Product Rule:

• Identify the numbers: Here, \( b = 5 \) and \( c = 2 \).
• Apply the property: Combine them inside a single log as \( \log (5 \times 2) \).

This step condenses the expression by transforming it into a single logarithmic entity, \( \log 10 \), which is easier to work with and simplifies calculations dramatically.

Logarithmic Expressions

Logarithmic expressions can often seem daunting, but by understanding the underlying rules and properties, these expressions become much more manageable. In the specific problem \( \log 5 + \log 2 \), the goal was to write this as a single logarithmic term using the properties of logarithms, particularly the Product Rule. This step:

• Simplifies calculations and algebraic manipulations.
• Helps in understanding the relationship between multiplication and addition in the logarithmic world.
• Makes finding solutions and evaluating the logs more straightforward without excessive computation.

By practicing these transformations, logarithmic expressions can be condensed efficiently, saving time and effort when tackling more complex mathematical problems.