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Chapter 3: Problem 35

Evaluate the logarithms exactly (if possible). $$\log _{2} 1$$

Short Answer

Expert verified
The exact value of \( \log_2(1) \) is 0.

Step by step solution

01

Understand the Logarithm Definition

The definition of a logarithm, \( y = \log_b(x) \), states that \(b^y = x\). Essentially, you are finding the power \(y\) to which the base \(b\) must be raised to produce the number \(x\).
02

Apply Definition to Given Problem

In our problem, we want to evaluate \( \log_2(1) \), which asks the question: what power must 2 be raised to, to get 1? Formally, if \(y = \log_2(1)\), then \(2^y = 1\).
03

Solve the Equation

Recognize that any number raised to the power of 0 is 1. Therefore, \(2^0 = 1\). So the value of \(y\) in the equation \(2^y = 1\) is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Definition
A logarithm is a mathematical operation that helps us determine how many times a base number must be multiplied by itself to reach a certain value. It works as an inverse operation to exponentiation. This relationship is essential to understand: if we say that a number equals a base raised to an exponent, the logarithm tells us the exponent. In simpler terms:
  • The logarithmic form: if given by \(y = \log_b(x)\), translates to the base \(b\) raised to the power of \(y\) equals \(x\).
  • For example, \(\log_2(8) = 3\) means that 2 must be raised to the power of 3 to get 8, i.e., \(2^3 = 8\).
This operation provides a way to transition between two different number scales鈥攐ne that's multiplicative (exponentiation) to one that's additive (logarithms). Understanding this conversion between multiplication and addition is key in various fields such as science and engineering.
Exponents
Exponents are the building blocks behind much of what we use in logarithms. An exponent tells you how many times to multiply a base number by itself. Written as \(b^y\), the base \(b\) is multiplied by itself \(y\) times.
  • For example, \(3^4\) means multiplying 3 by itself 4 times: \(3 \times 3 \times 3 \times 3\), which equals 81.
  • Another example, \(2^0 = 1\), illustrates the special case where any non-zero number raised to the power of zero results in 1.
These principles of exponents are crucial when evaluating logarithms, as they allow us to directly relate logarithmic expressions back to this fundamental concept of repeatedly multiplying a number by itself.
Evaluating Logarithms
Evaluating logarithms can initially seem challenging, but breaking the process down into a few clear steps can simplify it. Let's go through how to evaluate a basic logarithm using our example, \(\log_2(1)\):
  • First, use the logarithm definition to set up the equation: if \(y = \log_2(1)\), then we need to find \(y\) such that \(2^y = 1\).
  • By understanding our exponents, recognize that any base to the power of zero results in 1, thus \(2^0 = 1\).
  • Therefore, the evaluation of \(\log_2(1)\) gives us the result \(y = 0\).
By consistently applying the definition and the properties of exponents, you can navigate through any logarithmic challenge step by step. This approach ensures that even complex problems become manageable.

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Most popular questions from this chapter

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