Problem 21 - Elementary and Intermediate Algebra

Question

Evaluate expression. \(\log _{4} 4^{7}\)

Step-by-Step Solution

Answer

The value of \(\log_4(4^7)\) is 7.

Step 1: Apply Logarithm Property

Recall the logarithmic identity \[\log_b(x^n) = n \cdot \log_b(x)\]For this problem, identify the base and the exponent from \(\log_4(4^7)\). The base \(b\) is 4 and the exponent \(n\) is 7.

Step 2: Simplify the Expression

Substitute the values into the identity:\[\log_4(4^7) = 7 \cdot \log_4(4)\]Since the logarithm \(\log_4(4)\) equals 1, we can simplify further: \[7 \cdot 1 = 7\]

Key Concepts

Logarithmic Identities

Logarithmic identities play a vital role in simplifying and evaluating logarithmic expressions. These identities are rules that help us manipulate logarithmic functions, making them more relatable with algebraic expressions. One important identity often used in problem solving is the power rule for logarithms: - \[ \log_b(x^n) = n \cdot \log_b(x) \] This identity shows that you can bring down the exponent in front of the logarithm, simplifying the expression. It stems from the property of exponents in multiplication, allowing the transformation of power expressions into linear terms. - By utilizing these identities, tasks that seem complex at first can become much more manageable. The identity applied in the original problem effectively simplifies the expression from \( \log_4(4^7) \) to a straightforward multiplication problem.

Base of Logarithm

The base of a logarithm is a critical component that defines the scale or the reference point from which the logarithmic operation is performed. In the expression \( \log_4(4^7) \), the base is 4. This tells us that we are comparing powers of 4. - Understanding the base is crucial because logarithms essentially answer the question: "To what power must the base be raised, to produce the given number?" If the given number is already a power of the base, then evaluating the logarithm becomes much simpler using identities or basic properties. - When the number within the logarithm is expressed as a power of the base, we can use logarithmic identities to find the result efficiently. For example, in \( \log_4(4^7) \), knowing that 4 is the base and 4 raised to the power 1 gives 4, we can simplify computations very effectively.

Exponents in Logarithms

Exponents in logarithms communicate how many times a base is multiplied by itself to reach a certain value. In logarithmic expressions, recognizing and handling these exponents is essential to simplify and solve them. - Take the expression \( \log_4(4^7) \). Here, 7 is the exponent. Understanding exponents is key here because they indicate repeated multiplication: 4 multiplied by itself 7 times. - When dealing with logarithms, exponents can transform into coefficients, thanks to the power rule of logarithms: \( \log_b(x^n) = n \cdot \log_b(x) \). This rule helps us understand that an exponent outside the base logarithm can represent repeated multiplication efficiently through addition (since multiplying is essentially repeated addition). - In practice, this means that for \( \log_4(4^7) \), recognizing the exponent allows us to pull it out in front, simplifying the expression to a basic arithmetic operation.