Problem 21 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{3} 9 $$

Step-by-Step Solution

Answer

\(\log _{3} 9 = 2\)

Step 1: Understanding Logarithms

When you see a logarithm written like this: \(\log _{b} a = n\), it's the same as saying \(b^n = a\). From this, understand that a logarithm is asking: 'To what exponent must we raise the base b to obtain the number a?' In our given case, \(\log _{3} 9\), '3' is the base and '9' is the number whose logarithm to base '3' we need to find. So, we are being asked to find what power we must raise '3' to obtain '9'.

Step 2: Evaluating Logarithms

To evaluate \(\log _{3} 9\), you need to ask yourself, '3' to what power equals '9'? The answer is '2', because \(3^2 = 9\). This means \(\log _{3} 9 = 2\).

Step 3: Conclusion

By understanding the operation of logarithms, we found that \(\log _{3} 9 = 2\). This means we must raise '3' to the power of '2' to obtain '9'.

Key Concepts

Exponents

An exponent is a small number placed to the upper right of a base number. It tells you how many times to multiply the base by itself. For instance, if you see \(3^2\), it means you should multiply the number 3 by itself twice (3 * 3). In this context, the exponent is '2', which signifies that the base number, '3', should be squared.Exponents are crucial in understanding logarithms because a logarithm essentially asks the question: to what exponent do we raise a particular base to get a certain number? So when you see \(3^2 = 9\), you are seeing an exponent in action, demonstrating that multiplying the base '3', to itself once results in 9.Understanding exponents helps make the process of evaluating logarithms clearer. Write down different base-exponent combinations to see how they work together. This practice helps in retaining the concept.

Base

In mathematics, the base is the number that you are multiplying by itself when dealing with exponents. It serves as the foundation number in both exponential and logarithmic equations. For example, in the expression \(3^2\), '3' is the base.The base is crucial in logarithms, as it defines which number needs to be multiplied repeatedly to reach the desired outcome. In the problem \(\log_3 9\), the base is '3'. This means you need to consider how many times you must multiply '3' by itself to reach the number '9'.When dealing with logarithms, always identify the base correctly. It helps to determine how you manipulate the numbers in your calculations. Make sure to practice identifying the base in different logarithmic expressions to build your proficiency.

Evaluate

To evaluate a logarithmic expression means to find the exponent that corresponds to its given structure. In other words, you are finding what power the base must be raised to achieve the desired number. The evaluation process involves connecting the concepts of the base and the exponent together.For example, to evaluate \(\log_3 9\), you determine what power you must raise the base, 3, to get the number 9. By asking yourself, "3 to what power gives me 9?", and knowing that \(3^2 = 9\), you identify that the answer is 2. Therefore, \(\log_3 9 = 2\).The practice of evaluating logs is a great way to solidify your understanding of not just logarithms themselves but also more complex math such as exponential equations and calculus. Continually practice evaluating different logarithmic expressions to get comfortable with the process.