Problem 121 - Algebra and Trigonometry

Question

Explaining the Concepts. What question can be asked to help evaluate \(\log _{3} 81 ?\)

Step-by-Step Solution

Answer

The question 'What is the power to which 3 must be raised to obtain 81?' would aid in evaluating the expression \(\log _{3} 81\), and the answer will be 4.

Step 1: Understanding Logarithms

Logarithms have two parts: the base and the argument. In this case, the base is 3 and the argument is 81. Logarithms essentially answer the question: to what exponent do we need to raise the base in order to get the argument? In essence, if \(y = log_b(x)\), then it implies that \(b^y = x\).

Step 2: Apply the rule of Logarithms

We need to ask what power you need to raise 3 to get 81? Because \(3^4 = 81\), thus \(log_3(81) = 4\).

Step 3: Formulation of question to evaluate the expression

A good question to assist with this evaluation could be 'What is the power to which 3 must be raised to obtain 81?' This question directly leads to the evaluation step.

Key Concepts

Base And Argument

Understanding the base and argument is essential when working with logarithms.
When you look at a logarithm like \( \log_{3}(81) \), you have two main components:

• The base, which in this case is \(3\).
• The argument, which here is \(81\).

Logarithms are designed to answer a specific question. They tell us the exponent needed to obtain the argument from the base.
For example, \(| \log_b(x) = y | \) translates to \(| b^y = x | \).
Here, the base \(b\) is raised to the power of \(y\) to result in \(x\), the argument.
Think of it as asking: "To what power must the base be raised, to produce the argument?"
This knowledge will help you tackle these types of problems effectively.

Exponential Functions

Exponential functions play a vital role in understanding logarithms.
An exponential function can be represented as \( b^y\) where \(b\) is the base and \(y\) the exponent.
These functions model situations where growth or decay happens at a constant rate.

• Exponential growth involves a variable increasing at a consistent rate, like compound interest or population growth.
• Exponential decay refers to a variable decreasing steadily, like radioactive decay.

In the context of logarithms, understanding exponential functions allows us to reverse the process.
By knowing that \( 3^4 = 81 \) for example, you understand that the logarithmic equivalent \( \log_3(81) \) equals \(4\).
It's essentially the inverse operation. You find the power you need to reach a certain number starting from the base.
This interplay between exponential growth and logarithmic evaluation forms the backbone of understanding these mathematical concepts.

Logarithmic Evaluation

Logarithmic evaluation is the process of solving for the exponent in a logarithm.
This process can seem tricky, but it becomes manageable if you follow the correct steps.
Start by considering the question "What power should the base be raised to, to get the argument?"

• In \( \log_3(81)\), it asks us to find out: "What power do we need to raise \(3\) to, in order to get \(81\)?"
• The answer is \(4\) because \(3^4 = 81 \).

Evaluating logarithms often involves exploring exponential equivalents, so test different powers until you reach your argument.
Some questions can help to frame this process effectively:

• What are the simple powers of the base that might give you the argument?
• How can you manipulate these powers to reach a solution?

With practice, logarithmic evaluation becomes a straightforward task and is an essential mathematical skill.