Problem 4 - Algebra 2

Question

Write each equation in exponential form. \(\log _{3} 81=4\)

Step-by-Step Solution

Answer

The exponential form is \( 3^4 = 81 \).

Step 1: Understand the Logarithmic Form

The given equation is \( \log _{3} 81 = 4 \). This represents a logarithmic equation where 3 is the base of the logarithm, 81 is the result, and 4 is the exponent to which the base must be raised to yield the result.

Step 2: Convert to Exponential Form

To convert the logarithmic form to exponential form, recall that \( \log_b{a} = c \) can be rewritten as \( b^c = a \). In this case, \( \log_3{81} = 4 \) becomes \( 3^4 = 81 \).

Key Concepts

Exponential Form

Converting a logarithmic equation into its exponential form is a straightforward yet crucial concept in understanding logarithmic functions.
The exponential form gives us a clear view of what the logarithmic equation is expressing.
In the exponential form, the equation is structured as \( b^c = a \):

• \( b \) is the base.
• \( c \) is the exponent.
• \( a \) is the result.

For the equation \( \log_3{81} = 4 \), converting it to its exponential form, we express it as \( 3^4 = 81 \).
This form tells us that when 3 is raised to the power of 4, it equals 81.
Understanding this switch between forms enriches one's grasp of both the functions and how they relate to each other.

Logarithmic Equation

A logarithmic equation is a useful tool for solving problems involving exponential relationships.
The structure of a logarithmic equation is \( \log_b{a} = c \):

• \( b \) is the base of the logarithm, which is the number being repeatedly multiplied.
• \( a \) is the answer we would get from the exponential form.
• \( c \) is the power or exponent needed for the base to reach the result \( a \).

In our example, \( \log_3{81} = 4 \), we see that the base 3 needs to be raised to the power of 4 to produce 81.
This highlights the inverse relationship logarithms have with exponentials, allowing us to solve for unknowns in various mathematical contexts.

Base of a Logarithm

The base of a logarithm is a fundamental part of understanding how logarithmic functions operate.
The base is the number that is being raised to a power to result in another number and acts as a 'pivot' in handling exponential transformations.
In the equation \( \log_b{a} = c \):

• \( b \) is the base.
• These functions tell us how many times we need to multiply the base by itself to achieve a certain number.]

For instance, in \( \log_3{81} = 4 \), the base is 3, indicating how many times 3 must be multiplied together to result in 81.
This concept is vital as it forms the foundation for transitioning between logarithmic and exponential equations effectively.