Problem 41 - College Algebra Essentials

Question

Use properties of logarithms to condense 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

The expression \(\log 5+\log 2\) simplifies to \(\log 10\).

Step 1: Apply the property of logarithms

The expression given is \(\log 5+\log 2\). The property of logarithms states that the sum of two logarithms is equal to the logarithm of the product of their numbers. So, we can write the given expression as a single logarithm by multiplying the numbers 5 and 2.

Step 2: Calculate the value

Multiplying the numbers 5 and 2 gives a value of 10. Therefore, the expression \(\log 5+\log 2\) simplifies to \(\log 10\).

Key Concepts

Understanding the Properties of Logarithms

When working with logarithmic expressions, it's crucial to understand the basic properties of logarithms. These properties help simplify complex expressions, making calculations more manageable. One of the fundamental properties is the product property of logarithms. This states that the logarithm of a product is equal to the sum of the logarithms of its factors:

• \( \log_b (M \cdot N) = \log_b M + \log_b N \)

Here, \( b \) is the base of the logarithm, and \( M \) and \( N \) are the numbers we're working with. This property allows you to combine multiple logarithm expressions into a single one.
Besides the product property, there are other useful properties such as the quotient property and the power property. The quotient property states that the logarithm of a quotient is the difference of the logarithms. The power property tells us that the logarithm of a power is the exponent times the logarithm of the base. These properties are powerful tools that streamline solving logarithmic problems and make understanding exponential relationships easier.

The Process of Condensing Logarithms

Condensing logarithms is essential in simplifying expressions and solving equations. Condensing involves combining multiple logarithmic expressions into a single term. This is done by utilizing the properties of logarithms, particularly the product, quotient, and power properties.
For instance, given the expression \( \log 5 + \log 2 \), we can apply the product property to combine these two logarithms. Here’s how it works:

• The expression becomes \( \log(5 \times 2) \) by multiplying the numbers inside the logarithm.
• Thus, it condenses to \( \log 10 \).

The goal of condensing is to express the logarithm in its simplest form, often as a single logarithmic term with a coefficient of 1. This process is also helpful when solving equations that involve logarithms, as it reduces the complexity and makes finding solutions more straightforward.

Exploring Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms. They are used to solve equations that involve exponential growth and decay, making them crucial in fields like finance, physics, and biology. These expressions can often be simplified by applying the properties of logarithms.
A basic example like \( \log 5 + \log 2 \) illustrates the essence of working with logarithmic expressions. By understanding the context and the properties, such as the product property, you can simplify the expression easily to \( \log 10 \).
Logarithmic expressions can vary from simple to complex. Being able to condense and manipulate them, using properties like product, quotient, and power, allows you to solve problems more efficiently. In practice, this understanding aids in simplifying real-life problems involving rate changes, like population growth or chemical reactions, where logarithms help make sense of otherwise unmanageable numbers.