Problem 25 - Algebra 2 and Trigonometry

Question

Write each expression as a single logarithm. \(\log _{2} a+\log _{2} b\)

Step-by-Step Solution

Answer

\(\log_{2} (ab)\)

Step 1: Recognize the Logarithmic Addition Property

The expression \(\log_{2} a + \log_{2} b\) involves the sum of two logarithms with the same base, \(2\). According to the properties of logarithms, specifically the addition property, the sum of logarithms with the same base can be combined into a single logarithm.

Step 2: Apply the Addition Property of Logarithms

The addition property of logarithms states that \(\log_{b} x + \log_{b} y = \log_{b}(xy)\). In this problem, we apply the property as follows: \(\log_{2} a + \log_{2} b = \log_{2}(ab)\).

Step 3: Write the Expression as a Single Logarithm

Now, use the result from the previous step to express the original expression as a single logarithm. Thus, \(\log_{2} a + \log_{2} b\) simplifies to \(\log_{2} (ab)\).

Key Concepts

Addition Property of Logarithms

Understanding the addition property of logarithms is crucial for simplifying expressions where two or more logarithmic terms share the same base. When you see an expression like \( \log_b x + \log_b y \), it implies the addition of logarithms. The beauty of the logarithmic properties shines when this can be expressed as a single logarithm: \( \log_b (xy) \). This property is incredibly useful because it transforms an addition operation into a multiplication within the logs. This might initially seem surprising but remember, logarithms are all about exponents, and multiplication of arguments inside logarithms corresponds to adding exponents.
Understanding and utilizing this property efficiently allows for easier manipulation and calculation of larger expressions involving logs. So, the next time you see logs with the same base being added together, you know you can combine them into a single, more simplified logarithm.
Use this handy property whenever you spot that common log base, and you're looking at a sum of logarithmic expressions. It's a neat trick that saves time and simplifies your work.

Logarithmic Expressions

Logarithmic expressions are simply expressions that involve logarithms. At times, these expressions can appear daunting. However, they follow specific rules and properties that we can use to simplify them. In the context of our exercise, we had the expression \( \log_{2} a + \log_{2} b \).
Acknowledge that each logarithmic term has a similar structure: \( \log_b x \), where \( b \) is the base, and \( x \) is the argument. Here, the argument represents the "input" into the logarithm function.
By understanding each piece of the expression, you can manipulate it with ease. For instance, when multiple terms share the same base, as in our problem, properties like the addition property allow us to combine them into a single expression, which is less cumbersome to handle. This method reveals the power of logarithms to reduce complexity in mathematical expressions by managing large numbers through smaller exponents.
Keep an eye on the structure of logarithmic expressions, and you'll be able to transform them into simpler forms using the appropriate properties.

Base of Logarithm

The base of a logarithm is one of the fundamental components in logarithmic expressions. It determines the platform upon which the logarithm is built. Typically represented as \( b \) in \( \log_b x \), it informs how many times the base, \( b \), is multiplied to reach the "input," \( x \).
In our initial exercise, the base is \( 2 \), as seen in \( \log_{2} a + \log_{2} b \). Different bases can dramatically alter the meaning of a logarithmic expression. Most commonly, in various problems and textbooks, you'll encounter base \( 10 \) (common logs) and base \( e \) (natural logs), alongside custom bases like \( 2 \) in this example.
Estimating what base you're working with tells you the context within which the entire expression operates. All logarithmic properties, including the addition property used in our exercise, can only be effectively applied when dealing with terms that share the same base. This is key for ensuring you're conducting your operations correctly and efficiently.
Recognizing and correctly interpreting the base simplifies expressions and is fundamental to precise calculations in logarithmic math.