To solve the equation \( T(n) = \frac{3}{2} \log_2(n + 1) = 7.5 \) for \( n \), we will go through the following steps:

### Step 1: Isolate the logarithm
\[
\frac{3}{2} \log_2(n + 1) = 7.5
\]

### Step 2: Multiply both sides by \( \frac{2}{3} \) to isolate the logarithm
\[
\log_2(n + 1) = \frac{7.5 \times 2}{3} = 5
\]

### Step 3: Rewrite the equation in exponential form
The equation \( \log_2(x) = y \) is equivalent to \( x = 2^y \). Thus, we get:
\[
n + 1 = 2^5
\]

### Step 4: Solve for \( n \)
\[
n + 1 = 32
\]
\[
n = 31
\]

So, \( n = 31 \).

Would you like more details or have any questions? Here are five related questions:

1. How do you solve logarithmic equations involving different bases?
2. Can you explain how to convert between logarithmic and exponential forms?
3. How does changing the base of a logarithm affect the solution?
4. What is the significance of logarithms in algorithms and complexity analysis?
5. How do you apply logarithms to solve real-world problems?

**Tip:** Remember that the logarithm base 2, \( \log_2(x) \), is particularly common in computer science, especially in algorithms involving binary trees or binary search.

Learn how to solve the equation T(n) = 3/2 log2(n + 1) = 7.5 step-by-step with detailed explanations and calculations. This problem covers logarithms, exponential equations, and their application in algorithmic analysis. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation