Problem 26 - Algebra 2 and Trigonometry

Question

Write each expression as a single logarithm. 4 \(\log _{2}(x+2)\)

Step-by-Step Solution

Answer

\(4 \log_{2}(x+2)\) simplifies to \(\log_{2}((x+2)^4)\) as a single logarithm.

Step 1: Understanding the Problem

The exercise asks us to write the expression \(4 \log_{2}(x+2)\) as a single logarithm. We need to use the properties of logarithms to simplify this expression.

Step 2: Recall the Power Rule of Logarithms

The power rule of logarithms states that \(a \log_b(c) = \log_b(c^a)\). This rule allows you to bring a coefficient in front of a logarithm inside as an exponent.

Step 3: Apply the Power Rule

We apply the power rule to the expression \(4 \log_{2}(x+2)\). According to the rule, this becomes \(\log_{2}((x+2)^4)\).

Step 4: Write the Expression as a Single Logarithm

After applying the power rule, the expression \(4 \log_{2}(x+2)\) is rewritten as \(\log_{2}((x+2)^4)\). This is now a single logarithm.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is an incredibly useful property that helps in simplifying expressions involving logarithms. According to this rule, if you have a multiplier in front of a logarithm, such as in the expression \( a \log_b(c) \), you can bring that multiplier inside as an exponent. Thus, \( a \log_b(c) = \log_b(c^a) \).
This transformation is valuable because exponents, especially in algebraic terms, are often easier to work with than repeated multiplications. For example, when you encounter something like \( 4 \log_2(x+2) \), instead of multiplying \( \log_2(x+2) \) four times, you simply rewrite it as a single logarithm: \( \log_2((x+2)^4) \).
Understanding and applying this power rule makes handling logarithmic expressions more straightforward, simplifying the process of solving equations or altering expressions.

Logarithmic Expressions

Logarithmic expressions are mathematical phrases involving the logarithm function, often written through the notation \( \log_b(x) \), where \( b \) is the base and \( x \) is the argument. These expressions can appear in various forms and are used to represent exponentiation in a reversible way.
The core of understanding logarithmic expressions lies in appreciating how they relate to exponents. Essentially, the expression \( \log_b(x) \) asks the question: "To what power must \( b \) be raised to yield \( x \)?" For example, \( \log_2(8) \) equals 3, because \( 2^3 = 8 \).
When working with these expressions, it is essential to remember key properties, such as the power rule, the product rule \( \log_b(MN) = \log_b(M) + \log_b(N) \), and the quotient rule \( \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \). These properties help in manipulating and solving logarithmic equations effectively.

Simplifying Logarithms

Simplifying logarithms involves transforming complex or multi-part logarithmic expressions into a compact and more readable form, often with the aid of logarithmic laws. Simplifying allows for easier computation and enhances clarity.
In the provided exercise, you simplify \( 4 \log_2(x+2) \) by applying the power rule, transforming it into \( \log_2((x+2)^4) \). This technique reduces the clutter and combines multiple terms into one coherent expression, making it much easier to work with.
Other common simplification techniques include using the product, quotient, and change of base rules. For example, when given the expression \( \log_2(8) + \log_2(4) \), utilizing the product rule simplifies it to \( \log_2(32) \). Such processes not only reduce complexity but also enable quicker problem solving and verification of results.