Problem 4 - Intermediate Algebra

Question

For Problems \(1-10\), write each of the following in logarithmic form. For example, \(2^{3}=8\) becomes \(\log _{2} 8=3\) in logarithmic form. $$ 2^{6}=64 $$

Step-by-Step Solution

Answer

\(\log_2 64 = 6\)

Step 1: Identify the Base, Exponent, and Result

The given equation is in the exponential form, which is written as: \(2^6 = 64\). Here, 2 is the base, 6 is the exponent, and 64 is the result.

Step 2: Convert the Equation to Logarithmic Form

To convert the exponential form \(b^e = r\) to logarithmic form, it is rewritten as \(\log_b r = e\). In this problem, the base \(b = 2\), result \(r = 64\), and exponent \(e = 6\). Therefore, the logarithmic form is \(\log_2 64 = 6\).

Key Concepts

Exponential Form

Exponential form is a way to express numbers where a base number is raised to an exponent. It is written as \(b^e = r\). This structure is highly useful for simplifying large numbers or compactly expressing repeated multiplication. In our example, \(2^6 = 64\), here the base is 2 and it is multiplied by itself 6 times to get the result, 64.

• The base is the number being multiplied.
• The exponent signifies how many times the base is multiplied by itself.
• The result is the outcome of this multiplication.

Knowing exponential form allows you to understand and solve complex mathematical problems more easily. It is the foundation for comprehending logarithmic equations as well.

Base, Exponent, and Result

Understanding the components of an exponential expression is crucial. Let’s explore the parts: base, exponent, and result, using the expression \(2^6 = 64\).

• Base (2): This is the number being multiplied repeatedly. In our exercise, the base is 2. It serves as the core that structures the equation.
• Exponent (6): This number denotes the power to which the base is raised. It tells us how many times the base will be used in the multiplication. So, \(2^6\) means 2 multiplied by itself 6 times.
• Result (64): The resulting number from multiplying the base number as many times as indicated by the exponent. In this case, 2 multiplied 6 times gives us the result 64.

Grasping these elements makes it easier to transition between exponential and logarithmic forms.

Convert Exponential to Logarithmic

Converting an equation from exponential form to logarithmic form is a straightforward yet essential process in mathematics. The general rule for this conversion is to express \(b^e = r\) as \(\log_b r = e\).
To convert \(2^6 = 64\) into logarithmic form, we identify:

• Base (b): 2
• Exponent (e): 6
• Result (r): 64

By rearranging the exponential expression into its logarithmic counterpart, we write: \(\log_2 64 = 6\). This format answers the question: "To what power must the base 2 be raised, to yield 64?"
Understanding this conversion helps in solving problems that involve exponential growth or decay, common in scientific applications and real-world scenarios.