Problem 56 - Algebra 2

Question

Write each equation in exponential form. $$ \log _{6} 6=1 $$

Step-by-Step Solution

Answer

The exponential form of \(\log _{6} 6=1\) is \(6^1=6\).

Step 1: Identify the parts of the logarithmic equation

In a logarithmic equation, we have the base, the argument, and the result. For the given equation, \(\log _{6} 6=1\), we see that the base is 6, the argument is also 6, and the result is 1.

Step 2: Write the exponential form

The transformation of the logarithmic equation into the exponential form considers that the base raised to the result equals the argument. That is, in the form \( b^y=x \). Applying that to our equation, we have that 6 (the base), raised to 1 (the result of the logarithm), equals 6 (the argument). Thus the exponential form of \(\log _{6} 6=1\) is \(6^1=6\).

Key Concepts

Exponential Form

Exponential form is a way of expressing mathematical equations involving exponents. When you encounter a logarithmic equation and you need to convert it to an exponential form, think of it as changing the way the information is presented.
In general, if you have a logarithmic equation of the form \( \log_b x = y \), it can be rewritten as an exponential equation \( b^y = x \).
This tells us that \( b \), which is the base, raised to the power \( y \), gives you the argument \( x \). Looking at our specific example from the exercise, \( \log_6 6 = 1 \), this converts to the exponential form \( 6^1 = 6 \).
This shows that the base 6, raised to the power of 1, equals the argument, which is also 6.
Understanding this transformation is crucial because it allows you to see the relationship between logarithms and exponents, which is foundational in algebra and other areas of mathematics.

Logarithmic Equation

A logarithmic equation is one that involves the logarithm of a number. It is an equation that can express exponentiation in terms of logarithms. They often appear in mathematical problems and are key to solving a variety of exponential-related equations.
In a logarithmic equation like \( \log_b x = y \), the term \( \log \) signifies that you need to find the power \( y \) that the base \( b \) must be raised to, to produce the argument \( x \).
Consider the equation \( \log_6 6 = 1 \). Here, 6 is raised to the power of 1, which produces 6. Effectively, such equations help us solve for unknowns in algebra when dealing with exponential growth or decay, among other applications.
They are used in various fields including science and engineering, making them very versatile in real-world problem-solving.

Base and Argument

When dealing with logarithmic equations, it's important to understand the roles of the base and the argument. These are two fundamental components of a logarithmic expression.
The base, denoted as \( b \), is the number that is being raised to a power. For example, in \( \log_6 6 = 1 \), the base is 6. The base is the starting point for determining what power you need to achieve the argument.
Meanwhile, the argument, denoted as \( x \), is the result of the base raised to the specified power. In our exercise example, the argument is 6, as it is the number we must reach by raising the base to the required power.

• The base and argument combine to illustrate how logarithms relate to exponents.
• They serve to visualize the power relationship set in the logarithmic equation.

Understanding these components is essential for accurately converting between logarithmic and exponential forms, helping to solve equations and understand exponential growth.