Problem 26 - Elementary and Intermediate Algebra

Question

Evaluate expression. \(\log _{4} 4^{2}\)

Step-by-Step Solution

Answer

The expression \( \log _{4} 4^{2} \) evaluates to 2.

Step 1: Understanding the Logarithm Identity

The expression given is \( \log_{4} 4^{2} \). Using the logarithmic identity \( \log_b (b^x) = x \), where \( b \) is the base and \( x \) is the exponent, we can simplify the expression. In this case, the base \( b \) is 4, and \( x \) is 2.

Step 2: Applying the Logarithm Identity

Substitute the values from the expression into the logarithm identity. By applying \( \log_{4} (4^{2}) = 2 \), since the base of the logarithm \( 4 \) and the base of the exponent \( 4 \) are the same, the identity tells us that the expression equals the exponent 2.

Key Concepts

Logarithmic Identity

Logarithmic identities help us simplify expressions by recognizing common patterns. In mathematics, one powerful identity is the logarithmic identity. This is particularly useful when the base of the logarithm and the base of the exponent match. For example, the identity \( \log_b (b^x) = x \) states that if you have a logarithm \( \log_b \) and it is followed by its base raised to some exponent \( x \), the output is simply \( x \). This identity works because logarithms and exponents are inverse operations, much like addition and subtraction or multiplication and division.

• Use it to quickly simplify expressions where the base of the logarithm matches the base of the exponent.
• Notice that this identity doesn’t depend on the magnitude of \( b \); it only matters that both bases match.

Understanding this identity streamlines calculations and makes solving such expressions much easier.

Exponent

An exponent indicates how many times a number, known as the base, is multiplied by itself. In mathematical terms, when you see a number in the form of \( b^x \), "\( b \)" is the base and "\( x \) " is the exponent. The notation can be broken down to mean: multiply \( b \) by itself \( x \) number of times. The role of the exponent in our exercise is crucial, particularly because it appears in a logarithmic expression \( \log_{4} 4^2 \). In this context, the identity used relies upon the exponent's relationship with the base.

• Exponents can be thought of as repeated multiplication.
• Any number raised to the zero power is 1. For example: \( 4^0 = 1 \).
• An exponent of 1 means the number remains unchanged: \( b^1 = b \).

Understanding exponents is essential for breaking down and evaluating expressions, especially when applying logarithmic identities.

Base in Logarithms

The base in a logarithm is the number that gets raised to a power (the exponent) in the expression being evaluated. It's a foundational part of both the logarithm and the exponent expressions involved. In our problem \( \log_{4} 4^2 \), the number 4 serves as the base in both parts of the expression, making it easier to apply logarithmic identities.

• The base of the logarithm must be positive and not equal to 1.
• Changes in the base affect the evaluation of the logarithmic expression significantly.
• When a logarithmic expression has the same base in the numerical part and the logarithmic part, simplification becomes straightforward using identities.

Understanding the base is pivotal for interpreting logarithmic expressions, as it ultimately determines how the expression is simplified or evaluated.