Problem 42 - Algebra 2

Question

Evaluate each expression. \(\log _{4} 16^{x}\)

Step-by-Step Solution

Answer

The expression evaluates to \(2x\).

Step 1: Simplify the Expression Inside the Logarithm

Recognize that the expression inside the logarithm is a power of the base 4. In this case, note that 16 can be rewritten as 16 = 4^2. Therefore, substitute 16 in the expression: \[\log_{4}(16^x) = \log_{4}((4^2)^x)\]

Step 2: Apply the Power Rule for Exponents

Utilize the power rule for exponents, which states \((a^m)^n = a^{m \times n}\), to rewrite the expression:\[\log_{4}((4^2)^x) = \log_{4}(4^{2x})\]

Step 3: Use the Identity of Logarithms

Apply the identity \(\log_{b}(b^a) = a\), where the base of the logarithm and the base of the exponential part are the same:\[\log_{4}(4^{2x}) = 2x\]

Step 4: Final Evaluation

The evaluated expression is simply \(2x\), since all the operations have simplified the original problem to this term.

Key Concepts

Exponent Rules

Exponent rules play a crucial role in mathematics, especially when we deal with powers and roots. One integral exponent rule is the **Power Rule**:

• The Power Rule states \( (a^m)^n = a^{m \times n} \), meaning when you raise a power to a power, you multiply the exponents.
• This rule helps simplify expressions and is essential for managing exponential terms inside complicated expressions.

For example, in our original exercise, we use this rule to handle \( (4^2)^x \), which becomes \( 4^{2x} \). This process is vital in simplifying terms before applying logarithmic identities. Understanding exponent rules not only makes problems easier, but it also sets the stage for more proficient manipulation of complex expressions.

Logarithmic Identities

Logarithmic identities are as fundamental in logarithms as rules for arithmetic operations in basic math. A key identity is the **Identity of Logarithms**.

• It states \( \log_{b}(b^a) = a \), where \( b \) is the base of both the logarithm and the exponential expression.
• This identity simplifies many logarithmic calculations by converting them directly to their exponent.

In our exercise, after rewriting the expression as \( \log_{4}(4^{2x}) \) using exponent rules, we apply this identity to directly get the result \( 2x \). This step reduces the complexity of logarithmic operations, eliminating the base and leaving us with just the exponent, achieving simplification with remarkable efficiency.

Simplifying Expressions

Simplifying expressions involves breaking down complex terms into their simplest form. This concept plays an essential role when handling exponential and logarithmic expressions. Here is how you can approach it:

• First, recognize opportunities to express numbers as powers of other numbers, which assists in applying relevant rules.
• Incorporate exponent rules to reduce complicated power expressions.
• Utilize logarithmic identities to eliminate bases when exponent and log base are the same.

In practice, like in our example \( \log_{4}(16^x) \), we initially rewrite 16 as \( 4^2 \), enabling us to apply the power rule of exponents: \( 4^{2x} \). Finally, using the log identity \( \log_{4}(4^{2x}) = 2x \) allows for straightforward simplification. Simplifying expressions this way transforms them into easier parts, facilitating solving equations or evaluating expressions with minimal effort.