Question

Show that \(\log _{b} n=\Theta\left(\log _{2} n\right)\) for any constant \(b>1\).

Short answer

\(\log_b n = \Theta(\log_2 n)\) because they grow at the same rate, proportional by a constant.

Step-by-step solution

Understand the Logarithmic Functions

We need to show that the logarithmic function with base \(b\) and the logarithmic function with base \(2\) grow at the same rate. The notation \(\Theta\) represents an asymptotically tight bound, meaning that the two functions are asymptotically equivalent.

Use Change of Base Formula

Using the change of base formula for logarithms, we can express \(\log_b n\) in terms of \(\log_2 n\). The formula is: \(\log_b n = \frac{\log_2 n}{\log_2 b}\).

Express Relationship through Constants

The expression \(\log_b n = \frac{1}{\log_2 b} \log_2 n\) shows that \(\log_b n\) is proportional to \(\log_2 n\). This proportionality constant is \(\frac{1}{\log_2 b}\), which does not depend on \(n\).

Apply \\Theta-Notation

Substituting the expression from Step 3, we get: \(\log_b n = \Theta\left(\log_2 n\right)\). This follows because the constant multiplier \(\frac{1}{\log_2 b}\) ensures that the two functions grow at the same rate for large \(n\), adhering to the definition of \(\Theta\).

Key concepts

Logarithmic Functions

Logarithmic functions are fundamental in mathematics, particularly in topics involving exponential growth and decay. They are essentially the inverse of exponential functions. A logarithm answers the question: "To what power must the base be raised, to produce a given number?" For example, in the expression \(\log_b n = x\), \(b^x = n\).

Logarithmic functions can have different bases, with common ones being base 10 (common logarithm) and base \(e\) (natural logarithm). However, in computational and algorithmic contexts, using base 2 is quite prevalent, as it aligns with binary systems.

When comparing logarithms with different bases, it's important to note that they reflect similar growth characteristics, differing only by a constant multiplicative factor. This is key when discussing asymptotic behavior, as these functions can often be transformed into one another using the change of base formula.

Change of Base Formula

The change of base formula is an essential tool when working with logarithmic functions. It allows us to express a logarithm in one base in terms of another, which can simplify calculations and comparisons.

Given a logarithm \(\log_b n\), we can rewrite it using a different base, such as 2, using:

• \\log_b n = \frac{\log_2 n}{\log_2 b}\

This transformation is useful because it expresses the original logarithm as a multiple of another logarithm that might be easier to work with for a particular problem.

The constant \(\frac{1}{\log_2 b}\) plays a critical role in this formula. It serves as a proportionality constant that adjusts the function so that both logarithms can be compared fairly by accounting for the difference in their bases.

Asymptotic Equivalence

Asymptotic equivalence is a concept that helps simplify the analysis of functions by focusing on their behavior as the input grows large. Two functions are considered asymptotically equivalent if they grow at the same rate as their inputs become very large.

When we denote that \(f(n) = \Theta(g(n))\), it means \(f\) does not just approach \(g\) in terms of growth rate; it matches it tightly. Specifically, there exist positive constants \(c_1, c_2\), and \(n_0\) such that for all \(n \geq n_0\), \(c_1 \cdot g(n) \leq f(n) \leq c_2 \cdot g(n)\).

This is particularly relevant to logarithmic functions, where the change of base formula shows that any logarithmic function can be expressed as a constant multiple of another, regardless of their base. This directly leads to their classification as asymptotically equivalent, highlighting that despite base differences, they share a consistent growth pattern as \(n\) becomes large.