Problem 33 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \log _{2} 64 $$

Step-by-Step Solution

Answer

The simplified expression is 6.

Step 1: Understanding Logarithm Definition

The expression \( \log_{2} 64 \) asks us to find the power to which the base 2 must be raised to yield 64. In mathematical terms, this means finding \( x \) such that \( 2^x = 64 \).

Step 2: Express 64 as a Power of 2

64 is a power of 2. We need to express 64 as an exponent of 2. We try different powers of 2: \( 2^1 = 2 \), \( 2^2 = 4 \), \( 2^3 = 8 \), \( 2^4 = 16 \), \( 2^5 = 32 \), \( 2^6 = 64 \). Thus, we find that \( 64 = 2^6 \).

Step 3: Calculate the Logarithm

Using the result from Step 2, substitute \( 2^6 \) for 64 in the logarithm: \( \log_{2}(64) = \log_{2}(2^6) \). By the properties of logarithms, this simplifies to \( 6 \), because the logarithm function and the exponential function are inverses.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are written in the form \( b^x \), where \( b \) is the base and \( x \) is the exponent. In the context of our exercise, the base is \( 2 \) and we're interested in expressions like \( 2^x \), where \( x \) can take various values.

Key aspects of exponential functions include:

• The base must be a positive real number, except for 1.
• If the exponent is positive, the function grows rapidly as \( x \) increases.
• If the exponent is negative, the function approaches zero without becoming negative, as \( x \) decreases.

Exponential functions are important in many areas such as compound interest calculations, population growth models, and radioactive decay. Understanding them is essential for solving problems involving growth and decay.

Properties of Logarithms

Logarithms are the inverse of exponential functions. They provide a means to solve for unknown exponents in equations of the form \( b^x = y \). The expression \( \log_{b}(y) \) represents the exponent \( x \) that base \( b \) must be raised to, in order to produce \( y \). In the exercise, \( \log_{2}(64) \) tells us the power needed to raise \( 2 \) to get 64.

Here are some important properties of logarithms that can simplify calculations:

• \( \log_{b}(b^x) = x \): This tells us that the logarithm of a number raised to an exponent is the exponent itself.
• \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \): This property allows us to split the logarithm of a product into the sum of logarithms.
• \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y) \): This enables us to express the logarithm of a quotient as the difference of logarithms.
• \( \log_{b}(x^n) = n \cdot \log_{b}(x) \): This property allows the exponent to be brought out as a multiplier, simplifying calculations.

By utilizing these properties, complex logarithmic expressions can often be rewritten and solved much more easily.

Powers of 2

Powers of 2 are ubiquitous in both mathematics and computer science. They follow a simple pattern where each successive power is double the previous one. Understanding these is critical when dealing with binary systems and logarithms that use base 2.

The sequence of powers of 2 begins with:

• \( 2^0 = 1 \)
• \( 2^1 = 2 \)
• \( 2^2 = 4 \)
• \( 2^3 = 8 \)
• \( 2^4 = 16 \)
• \( 2^5 = 32 \)
• \( 2^6 = 64 \)

Noticing the pattern can assist in quickly identifying which power of 2 equals a given number. As we solve \( \log_{2}(64) \), we recognize that 64 can be expressed as \( 2^6 \). Therefore, knowing powers of 2 helps simplify such logarithmic expressions effectively.