Question

Which is different? Find "both" answers. What power of 4 gives you 16 ? What is log base 4 of 16 ? Evaluate \(4^2\) Evaluate \(\log _4 16\)

Short answer

For the questions 'What power of 4 gives you 16?' and 'What is log base 4 of 16?', the answer is 2. For the questions 'Evaluate \(4^2\)' and 'Evaluate \(\log _4 16\)', the answers are 16 and 2, respectively. Thus, the different one is ‘Evaluate \(4^2\)' as its answer is not 2, but 16.

Step-by-step solution

Power of 4 that equals 16

The exponent that when applied to the base number 4 results in 16 is 2. This can be written as: \(4^2 = 16\). So, the answer here is 2.

Log base 4 of 16

The base-4 logarithm of 16 is 2. Logarithm is the exponent to which the base 4 must be raised to get 16. This can be written as: \(\log _4 16 =2 \). The result here is also 2.

Evaluate \(4^2\)

The number 4 raised to the power of 2 equals 16, so \(4^2 =16 \). Here the answer is 16.

Evaluate \(\log _4 16\)

The base-4 logarithm of 16 is 2. As discussed before, the logarithm finds the number we need to raise our base (4) to get another number (16), so \(\log _4 16 = 2 \). The result here is also 2.

Key concepts

Exponents

Understanding exponents is essential to mastering many areas of mathematics, including algebra and calculus. An exponent refers to the number of times a base number is multiplied by itself. For example, when you see an expression like \(4^2\), the base number is 4 and the exponent is 2. This indicates that 4 is to be multiplied by itself once, resulting in \(4 \times 4 = 16\).

When evaluating powers, remember that the base can be any real number, and the exponent signifies how many times to use the base in multiplication. Exponents can also be negative, which means you need to divide by the base rather than multiplying it, or zero, which always results in the number one. A clear grasp of exponents provides a solid foundation for understanding more complex mathematical concepts.

Base-4 Logarithm

A logarithm is an operation that is essentially the inverse of exponentiation. In a base-4 logarithm, the base of the logarithm is 4. The expression \(\log_4 16\) asks the question: to what power must 4 be raised to produce 16? The answer, as given in the problem, is 2, because \(4^2 = 16\).

It's key to remember that the base of the logarithm must always be greater than 0 and cannot equal 1. Logarithms are extremely useful in many fields such as engineering, sciences, and finance, due to their ability to simplify the multiplication and division of large numbers into more manageable addition and subtraction operations. Understanding how to evaluate logarithms can deepen your comprehension of exponential relationships.

Evaluating Powers

Evaluating powers is a core skill in mathematics, where you determine the result of a base number raised to an exponent. This is a straightforward process when you’re dealing with small numbers or integers. For instance, evaluating \(4^2\) involves multiplying 4 by itself to get 16. However, when exponents are fractions, negative numbers, or involve roots, the process can get more intricate.

Knowing how to evaluate powers also allows you to understand growth patterns, decay in science, and financial calculations involving interest. It is essential to practice with various exponents to gain fluency in recognizing patterns and applying the rules of exponents to solve them effectively. Evaluating powers is not just about the simple calculation but understanding the rate of change and scaling which is crucial in many real-world applications.