Problem 17 - Precalculus Functions and Graphs

Question

Find the indicated value of the logarithmic functions. $$\log _{2}(64)$$

Step-by-Step Solution

Answer

The value of \(\text{log}_{2}(64)\) is 6.

Step 1: Identify the Logarithmic Function

The problem asks to find the value of \(\text{log}_{2}(64)\). This means we need to determine what power of 2 equals 64.

Step 2: Convert to Exponential Form

Rewrite the logarithmic equation \(\text{log}_{2}(64)\) as an exponential equation: \(2^x = 64\).

Step 3: Solve the Exponential Equation

Determine the value of \(x\) for which \(2^x = 64\). Notice that \(64 = 2^6\). Therefore, \(x = 6\).

Step 4: Conclude the Result

Since \(\text{log}_{2}(64) = 6\), the indicated value of the logarithmic function is 6.

Key Concepts

Logarithms

A logarithm answers the question: For a given number, what power must another fixed number (the base) be raised to in order to produce that given number? For example, in \(\text{log}_{2}(64)\), we are trying to find out how many times we need to multiply 2 by itself to get 64. This 'how many times' is the exponent. So, if we say \(\text{log}_{b}(a) = c\), it means \({b^c = a}\). We usually write this as \(\text{log}_{b}(a) = c\). With this in mind, let’s explore core aspects of logarithms through different lenses.

Exponential Equations

Exponential equations involve expressions where variables appear as exponents. They have the form \({b^x = a}\). We often convert logarithmic equations into exponential equations to make them easier to solve. In our example, \(\text{log}_{2}(64)\), we converted it to \({2^x = 64}\). This is a straightforward process that makes solving for the exponent more intuitive. To solve \({2^x = 64}\), we need to recognize patterns or use known exponent rules. Since \({2^6 = 64}\), we see that \({x = 6}\). This type of conversion is crucial in simplifying both the process and understanding of equations.

Base 2 Logarithm

The base of a logarithm is critical to solving the equation correctly. In the case of \(\text{log}_{2}(64)\), the base is 2. This means we are dealing with powers of 2. The base 2 logarithm is particularly common in computer science and information theory because it deals with binary numbers. In general, base 2 logarithms help us understand how many times a number must be divided by 2 to get to 1. This is useful in algorithms and computational complexity. So, \(\text{log}_{2}(64) = 6\) tells us that 2 raised to the power of 6 equals 64.

Logarithmic to Exponential Conversion

The conversion from logarithmic form to exponential form is a key skill in solving logarithmic equations. To convert \(\text{log}_{b}(a) = c\) to exponential form, you rewrite it as \({b^c = a}\). This conversion allows us to deal with the equation in a more familiar form. For our problem \(\text{log}_{2}(64) = x\), we write it in exponential form as \({2^x = 64}\). By doing this, we leverage our understanding of exponents to solve the equation. Recognizing that \({2^6 = 64}\) immediately gives us that \({x = 6}\). This method enhances comprehension and enables efficient problem-solving.