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Chapter 9: Problem 13

Express as an equivalent expression that is a single logarithm. $$\log _{a} 2+\log _{a} 10$$

Short Answer

Expert verified
\( \log_{a} 20 \)

Step by step solution

01

Identify the Logarithm Property

Recognize the logarithm property that combines the sum of two logarithms into a single logarithm. The property is \( \log_b(x) + \log_b(y) = \log_b(xy) \).
02

Apply the Property

Using the property identified, combine the logarithms: \( \log_{a} 2 + \log_{a} 10 = \log_{a} (2 \times 10) \).
03

Simplify the Expression

Multiply the numbers inside the logarithm: \( 2 \times 10 = 20 \). Therefore, \( \log_{a} (2 \times 10) = \log_{a} 20 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Logarithms
Combining logarithms is a critical concept in algebra. It helps to simplify complex expressions and solve equations effectively. The primary property you use for combining logarithms is the **Product Rule for Logarithms**. This rule states that the sum of two logarithms with the same base is equivalent to the logarithm of the product of their arguments. The formula is \( \log_b(x) + \log_b(y) = \log_b(xy) \). For example, combining \( \log_a 2 + \log_a 10\) using this rule leads to \( \log_a(2 \times 10) \). Simplifying inside the logarithm gives \( \log_a 20 \). This method helps reduce multiple logarithms into a single, more manageable expression.
Equivalent Expressions
An equivalent expression maintains the value of the original expression but appears in a different form. In logarithms, converting a sum of logs into a single log is an example of creating an equivalent expression. For instance, \( \log_a 2 + \log_a 10\) seems complex, but by using the Product Rule to combine them, we simplify it to \( \log_a 20 \). This new expression is equivalent to the original; it represents the same quantity but is easier to work with. This skill is pivotal not just for simplification but also for solving more intricate logarithmic equations.
Logarithmic Equations
Logarithmic equations are equations that involve logarithms with variables inside them. To solve these, you often need to combine and simplify logarithmic expressions. One common approach is to use the properties of logarithms, such as the Product Rule, to merge multiple logs into one. For example, an equation containing \( \log_a 2 + \log_a 10 = c \) can be simplified to \( \log_a 20 = c \). This reduction helps to isolate the variable and solve the equation. Understanding how to combine logarithms and create equivalent expressions is thus crucial for effectively tackling logarithmic equations.

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