Problem 17 - Algebra A Combined Function

Question

Write each as a logarithmic equation. $$ 2^{4}=16 $$

Step-by-Step Solution

Answer

\( \log_2(16) = 4 \)

Step 1: Understanding the components

A logarithmic equation is written as \( \log_b(a) = c \), where \( b \) is the base, \( a \) is the result, and \( c \) is the exponent. In the expression \( 2^4 = 16 \), our base \( b \) is 2, the value \( a \) is 16, and the exponent \( c \) is 4.

Step 2: Applying the logarithmic form

Convert the expression \( 2^4 = 16 \) into a logarithmic equation using the identified components—base, result, and exponent. This gives us \( \log_2(16) = 4 \).

Step 3: Conclusion: Final Logarithmic Equation

After rewriting \( 2^4 = 16 \) as a logarithmic equation, we have the final result: \( \log_2(16) = 4 \).

Key Concepts

Understanding Logarithms

Logarithms are mathematical expressions that help us understand the power to which a number, called the base, is raised to produce a given number. Essentially, a logarithm answers the question: "To what exponent must the base be raised to equal a certain number?"
For example, in the logarithmic expression \( \log_b(a) = c \), \( b \) is the base, \( a \) is the result, and \( c \) is the exponent. This means "\( b \) raised to the power of \( c \) equals \( a \)."
Logarithms are incredibly useful in various fields such as science, engineering, and finance for simplifying complex equations and solving exponential growth problems. They also help in converting multiplication into addition, which can simplify calculations significantly.

• In the context of our exercise, the logarithm helps translate the exponential expression \( 2^4 = 16 \) into \( \log_2(16) = 4 \).
• It tells us that the base 2 must be raised to the power of 4 to result in 16.

From Exponential to Logarithmic Form

The process of converting an exponential expression into a logarithmic one can be simple once you understand the relationship between the two.
An exponential form such as \( b^c = a \) directly translates to a logarithmic form \( \log_b(a) = c \).
In our example:

• The exponential equation \( 2^4 = 16 \) tells us that 2, the base, to the power of 4, equals 16.
• By converting it to a logarithmic equation, \( \log_2(16) = 4 \), the equation states that the logarithm of 16 with base 2 is 4.
• This transformation makes it easier for us to solve problems where you're trying to find out the exponent.

This switch from exponential to logarithmic forms is crucial in simplifying mathematical problems. It allows easy manipulation of equations by turning exponents into logarithms, thus aiding in the solving process.

Base and Exponent Explained

In any exponential or logarithmic expression, understanding the roles of the base and exponent is key.
The base in an expression \( b^c = a \) is the number that gets multiplied by itself. The exponent is the power to which the base is raised. It tells how many times the base is used as a factor in the multiplication.
Analyzing our equation \( 2^4 = 16 \):

• "2" is the base. It is the number being multiplied.
• "4" is the exponent. It shows that we multiply 2 by itself four times: \( 2 \times 2 \times 2 \times 2 \).
• The product is 16, which is the result of this operation.

Understanding these components helps immensely in working with both exponential and logarithmic equations. The base and the exponent are the building blocks that define the entire equation, leading to the solution of the logarithmic operation \( \log_2(16) = 4 \). They are fundamental in understanding how numbers behave under different operations, especially when transitioning from exponential to logarithmic forms.