Question

Rewrite the equation in exponential form. (See Example 1.) \(\log _6 1=0\)

Short answer

The exponential form of the given logarithmic equation \(\log _6 1 = 0\) is \(6^0 = 1\).

Step-by-step solution

Understanding Logarithmic Properties

It's important to remember the basic rule of logarithms, which is that \(\log_a b = c\) is equivalent to \(a^c = b\). Here, 'a' is the base of the logarithm, 'b' is the argument of the logarithm, and 'c' is the value of the logarithm.

Identifying the Logarithm Elements

In the given problem, \(\log _6 1 = 0\), '6' is the base 'a', '1' is the argument 'b' and '0' is the value 'c'.

Convert the Logarithm into Exponential Form

After identifying the individual elements from the given logarithmic equation, the conversion into exponential form can be done using the basic equivalency rule mentioned in step 1. Therefore, the equation becomes \(6^0 = 1\).

Key concepts

Logarithms

Logarithms are a fundamental concept in mathematics, often used to simplify complex calculations and solve equations. They tell us the power to which a number, known as the base, must be raised to obtain another number. Think of them as the opposite of exponential functions.
For example, in the equation \(\log_a b = c\), the logarithm tells us that the base \(a\) raised to the power of \(c\) equals \(b\). This is central because it allows us to move between different forms of equations easily.
Logarithms have several practical uses, such as in measuring the intensity of earthquakes (Richter scale), sound (decibels), and even in computing algorithms. Always remember, the base of a logarithm must be greater than zero, and it can never be one since it leads to undefined scenarios.

Base of the Logarithm

The base of a logarithm is a critical component in the logarithmic equation. It is the number that is raised to a power, determined by the logarithm, to equal the argument of the logarithm. In the expression \(\log_a b = c\), 'a' represents the base.
Choosing the right base depends on the problem you're dealing with. Common bases include 10 (common logarithm), \(e\) (natural logarithm), and 2 (used frequently in computer science).
In our example, we consider the logarithm \(\log_6 1 = 0\). Here '6' is chosen as the base, which helps us to convert this logarithmic equation smoothly into its exponential form. Understanding the role of the base is essential in solving logarithmic equations and accurately switching between forms.

Exponential Equations

Exponential equations are the counterpart of logarithmic equations. They involve expressions where a base is raised to a variable exponent. Converting a logarithmic equation to an exponential equation often simplifies the process of finding solutions.
For our problem, converting \(\log _6 1 = 0\) into exponential form yields the equation \(6^0 = 1\). This exercise demonstrates that any number, regardless of its value, raised to the power of zero, equals one.
Solving exponential equations typically involves expressing both sides of the equation as powers with the same base and equating the exponents. Understanding both logarithmic and exponential forms and how to move between them is pivotal in addressing a broad range of mathematical problems.