91Ó°ÊÓ

Logarithms can be thought of as the inverse functions of exponentiation. A logarithm answers the question: 'To what exponent must a given base be raised, to produce a certain number?' We denote logarithms as \(\text{log}_b (a)\), where \(b\) is the base and \(a\) is the number we want to achieve through exponentiation. In simpler terms, if we have \(b^x = a\), then \(x = \text{log}_b (a)\).
For the exercise given, you need to understand that \( \text{log}_3 (1) \) asks, 'To what power must 3 be raised to get 1?' Since any number raised to the power of 0 equals 1, \( \text{log}_3 (1) \) is 0. This fundamental concept is key to unlocking more complex problems involving logarithms.

An exponential equation is one in which variables appear as exponents. For example, \(3^x = 1\) is an exponential equation. To solve this type of equation, we often use logarithms.
Here's how it applies to our exercise:

• We start by writing the logarithmic equation in its exponential form.
• For \( \text{log}_3 (1) \), we write \(3^x = 1\).
• We need to find the value of \(x\) that makes this equation true.

Since any number raised to the power of 0 is 1, we find that \( x = 0\). Thus, \( \text{log}_3 (1) \) equals 0. Understanding how to convert between exponential and logarithmic forms is crucial for solving these equations.

In the context of logarithms and exponentiation, the base and exponent are fundamental components. The base is the number that is being raised to a power. The exponent specifies how many times the base is multiplied by itself.
When we look at \( \text{log}_3 (1) \), the base is 3, and we need to find the exponent that makes 3 raised to this exponent equal to 1.
A few tips to remember about base and exponent:

• The base (like our 3) is always positive in our problems.
• The exponent (what we're solving for) can be positive, negative, or zero.
• Any base raised to the power of zero is always 1.
• Other exponents define repeated multiplication (for positive exponents) or division (for negative exponents).

This understanding lets us see clearly why \( \text{log}_3 (1) = 0 \), because \(3^0 = 1\). This matching of base and exponent pairs is a key skill for solving logarithmic functions.