Question

Evaluate the expression without using a calculator.\(\log _{3} 9\)

Short answer

The expression \(\log _{3} 9\) evaluates to 2.

Step-by-step solution

Understanding Logarithms

A logarithm is an operation that determines the number of times a certain number, known as a base, is multiplied by itself to reach another number. In the expression \(\log _{3} 9\), 3 is the base, and the aim is to find the exponent, or the number of times 3 is multiplied by itself to reach 9.

Evaluating the Expression

Remembering that \(\log _{b} (b^{x}) = x\), it's clear that since \(3^2 = 9\), the equation \(\log _{3} 9\) evaluates to 2.

Key concepts

Understanding the Logarithm Base

In the journey of mastering logarithms, grasping the concept of the 'logarithm base' is a pivotal first step. Imagine a logarithm as a question in the form of an equation. It asks, 'To what power must I raise the base to obtain a certain number?' For instance, when navigating the expression \(\log _{3} 9\), the number 3 is what we call the 'logarithm base'. Here, the base is like the foundation of the operations: we are repeatedly multiplying 3 by itself to reach our target number, which in this case is 9.

Contextualizing this with a real-life metaphor, if you think of it as a game of stacking blocks, the base block is the kind of block you are using, and you're trying to figure out how many of these blocks stacked on top of each other will reach a certain height. Each stacking represents multiplying the number by itself, and when the structure matches the desired height (our target number), the count of blocks used (the exponent) is the answer to our logarithmic query.

Demystifying Logarithmic Expressions

Expanding on our understanding of logarithms, we must also unravel the mystery of 'logarithmic expressions'. These expressions, such as \(\log _{3} 9\), encapsulate the entire logarithmic operation and serve as a concise mathematical way of asking the question we talked about earlier.

Here's a friendly tip: whenever you see a logarithmic expression, you can translate it in your head to a simpler form by saying '3 to what power equals 9?' This mental translation is often the key to unlocking the problem without confusion. Delving into the specifics, our expression has two main components: the base (which we discussed in the previous section) and the argument, which in our example is 9. The expression \(\log_{base} (argument)\) is the standard format, and understanding it means you're well on your way to becoming proficient in the world of logs.

Exponents: The Power Behind the Logarithms

Now, let's turn our attention to 'exponents.' You've encountered these in various mathematical scenarios, from squaring a number to calculating compound interest. In essence, an exponent tells us how many times to multiply a number by itself. For instance, the expression \(3^2\) indicates that 3 should be multiplied by itself exactly two times, giving us 9.

This relationship between base and exponent is crucial when evaluating logarithms. In our example, the exponent is the answer we seek when we evaluate \(\log _{3} 9\). Since we know that \(3^2 = 9\), the exponent is 2, making it the solution to our logarithmic expression. Remember this connection, and you'll be able to tackle a myriad of logarithmic problems with confidence.