Problem 21 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{3} 9$$

Step-by-Step Solution

Answer

The value of \( \log_{3} 9 \) is 2.

Step 1: Understand the properties of logarithms

The logarithm of a number in a certain base is the power or exponent to which the base must be raised to get that number. In other words, if \( b^y = x \), then \( \log_b x = y \). So in the expression \( \log_{3} 9 \), we need to find an exponent such that \( 3^{exponent} = 9 \).

Step 2: Calculate the logarithm

We know that \( 3^2 = 9 \), so the exponent is 2. Therefore, \( \log_{3} 9 = 2 \).

Key Concepts

Logarithmic Expression

When we talk about logarithmic expressions, we are dealing with an alternative way of expressing exponentiation. The logarithm tells us what power we need to raise a base to achieve a certain number. For example, the logarithmic expression \( \log_{3} 9 \) asks the question: 'To what power must we raise 3 to get 9?'

Understanding logarithms is crucial because they convert multiplicative processes into additive ones, which is handy for simplifying complex mathematical operations. So, in every logarithmic expression, we have three parts: the base (here, it's 3), the logarithm of a number (which is 9 in this case), and the answer to the expression, which is the exponent.

Here's how we interpret it: 'The logarithm base 3 of 9 is equal to what power?' The answer, as we find in our exercise, is 2. This relationship between the base, the logarithm, and the exponent makes understanding logarithms a fundamental skill in various fields of study.

Calculating Logarithms Without a Calculator

Calculating logarithms doesn't always require a calculator, especially when the numbers involved are friendly or part of common exponential pairs. A solid grasp of exponent rules and familiarity with squares, cubes, and higher powers of small numbers can go a long way.

Here's a simple approach to calculating a logarithm without a calculator: First, identify if the number you are taking the log of is a power of the base. If it is, your job is done, as the exponent is your answer. If not, try to express the number as a product or quotient of powers of the base, or use logarithm properties to simplify the expression.

It's much like being a detective; you look for clues (factors and multiples related to the base), piece them together, and solve the mystery (find the exponent). While not all logarithmic values can be found this neatly, many used in educational exercises are designed for straightforward calculation.

Exponent and Base Relationship

The relationship between the exponent and the base in logarithms is pretty straightforward: the base raised to the power of the exponent gives you the number whose logarithm you're trying to find. In the given exercise \( \log_{3} 9 \) we can see this relationship clearly. We know that \( 3^2 = 9 \), so the base (3) raised to the power (2) equals the number (9).

This relationship forms the backbone of how we interpret logarithms. When we see \( \log_{b} x = y \), we must understand that \( b^y = x \). This concept is so fundamental that it's echoed across various logarithmic properties and applications. In finance, for example, the rule of 72 uses this relationship to estimate investment doubling time, while in science, the pH scale measures the acidity or basicity of a solution using the negative of the base 10 logarithm.