Problem 29 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression without using a calculator.\(\log _{2} \frac{1}{16}\)

Step-by-Step Solution

Answer

-4

Step 1 - Relationship between log base and number

The first step would be to express 1/16 in terms of 2. We can easily say that \( \frac{1}{16} = 2^{-4} \) as \( 2^4 = 16 \) and 1 divided by 16 is the same as 2 raised to the power of -4.

Step 2 - Plugging into the logarithm

Now we plug this into the logarithm which now gets simplified as: \( \log _{2} 2^{-4} \)

Step 3 - Logarithmic rule

We know that the base 2 logarithm of 2 to any power is just the power itself. So, the base 2 logarithm of \( 2^{-4} \) is \( -4 \)

Key Concepts

Logarithms

When solving a logarithm, you're finding the exponent or power to which a base number, such as 2, must be raised to obtain another number. A logarithm tells you how many times a base number is multiplied by itself to reach a certain value. For example,

• \( \log_{2} 8 = 3 \) because you need to raise 2 to the power of 3 to get 8 (\( 2^3 = 8 \)).
• Similarly, \( \log_{2} 2^{-4} = -4 \) because 2 raised to the power of -4 gives \( \frac{1}{16} \).

Understanding logarithms involves grasping various properties, such as the change of base formula, product, quotient, and power rule of logarithms. For this exercise, the power rule simplifies our calculation radically, turning a complex expression into a simple integer or fraction.

Powers of Two

In many mathematical contexts, especially in computer science, powers of two play a critical role. Powers of two are integers that can be written as \( 2^n \), where \( n \) is an integer.

• Basic powers include values like: \( 2^0=1 \), \( 2^1=2 \), \( 2^2=4 \), \( 2^3=8 \), and \( 2^4=16 \).
• They extend towards negative exponents as well, such as \( 2^{-1}=\frac{1}{2} \), \( 2^{-2}=\frac{1}{4} \), and \( 2^{-4}=\frac{1}{16} \).

These expressions demonstrate how powers of two can represent fractions, which can be very useful. When you know the powers of two by memory, evaluating logarithmic expressions becomes straightforward as seen in this exercise.

Negative Exponents

Understanding negative exponents clarifies why fraction representations occur. A negative exponent essentially means taking the reciprocal of the number with the positive exponent. This changes large powers-based numbers into fractions easily.

• For example, \( 2^{-3} \) is the same as \( \frac{1}{2^3} \), which equals \( \frac{1}{8} \).
• In this exercise, \( 2^{-4} \) equates to \( \frac{1}{2^4} \) making it \( \frac{1}{16} \).

Whenever you see a negative exponent, think of it as flipping the base number and raising it to a positive power. Mastery of negative exponents ensures your ability to transform and simplify expressions like the ones involving logarithms of fractional numbers seamlessly.