Question

Evaluate the logarithm. \(\log _3 3\)

Short answer

The logarithm \(\log_3 3 = 1\).

Step-by-step solution

Understand the Logarithm

A logarithm is the inverse to exponentiation. That means the logarithm of a given number \(x\) is the exponent to which another fixed number, the base \(b\), must be raised to produce that number \(x\). In the logarithm \(\log _b x = n\), the number \(b\) is the base, \(n\) is the exponent and \(x\) is the number the base must be raised to get. In other words, \(b^n = x\).

Apply the Principle of Logarithm Base-equality

When the number \(x\) is the same as the base \(b\) (\(b = x\)), the logarithm evaluates to 1, because any number raised to the power of 1 is itself. Think about this expression: \(b^1 = b\), which in logarithm form is \(\log_b b = 1\).

Evaluate the Given Logarithm

Applying this principle to the given problem \(\log_3 3\), it's clear that the base of the logarithm (3) is equal to the number itself (3). Therefore, \(\log_3 3 = 1\).

Key concepts

Logarithmic functions

Imagine a logarithmic function as a detective in a mystery novel, whose sole mission is to uncover the hidden exponent in a number's story. The logarithm does this by asking: 'To which power do we need to raise the base in order to get the target number?'

Specifically, in the mathematical expression \( \log_b x = n \), think of \( x \) as the 'target number', \( b \) as the 'base', and \( n \) as the 'mystery exponent'. For example, if our detective-logarithm gets the case of \( \log_3 27 = ? \), it quickly deduces that 3 raised to the power of 3 equals 27, therefore the mystery exponent \( n \) is 3.

A logarithmic function always ties back to this core pattern. It's crucial for students to visualize this relationship to understand how the function operates across different equations.

Exponents and logarithms

The dynamic duo of mathematics, exponents and logarithms, are two sides of the same coin. Where an exponent says \( b^n = x \), its alter ego the logarithm responds \( \log_b x = n \). To switch between these identities, you simply need to flip the script.

An exponent involves taking a base \( b \) and raising it to a power \( n \) to achieve the result \( x \), while a logarithm works in the opposite direction to identify what that power \( n \) is, given \( b \) and \( x \).

The Exponential Question

If you have \( 2^3 = 8 \), you're observing an exponential function at work, declaring that 2 to the third power equals 8.

The Logarithmic Retort

In contrast, the logarithm looks at \( 8 \) and the base \( 2 \) and calculates backward, concluding \( \log_2 8 = 3 \). This intimate relationship is foundational for students to master other advanced concepts in mathematics.

Properties of logarithms

The properties of logarithms are the set of rules that explain how these mathematical detectives play by the rules in their quest to find missing exponents. One such fundamental property is the base-equality property that we applied in the textbook exercise.

The base-equality property states that \( \log_b b = 1 \), since any base \( b \) raised to the 1st power is itself. This is a cornerstone property because it applies universally across all bases, acting almost like a mathematical axiom.

But that's not all. There are other properties, such as the product property (which allows logarithms to multiply), quotient property (dividing within logarithms), and the power property (dealing with exponents within logarithms). Each of these properties helps in breaking down and simplifying logarithmic expressions, enabling students to navigate complex logarithmic problems with ease.