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Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
Chi-Squares Test Bivariate Analysis on Categorical Categorical Variables
Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
Supervised Learning vs Unsupervised Learning Reinforcement Learning Generative and Descriminative Models Parametric and Non Parametric model
Machine Learning Pipeline Preparing Dataset Build a Benchmark Model: Regression Build a Benchmark Model: Classification
Evaluation Metrics for Machine Learning Everyone should know Confusion Matrix Accuracy Precision and Recall AUC-ROC Log Loss R2 and Adjusted R2
Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Understanding Cost Function Understanding Gradient Descent Math Behind Gradient Descent Assumptions of Linear Regression Implement Linear Regression from Scratch Train Linear Regression in Python Implementing Linear Regression in R Diagnosing Residual Plots in Linear Regression Models Generalized Linear Models Introduction to Logistic Regression Odds Ratio Implementing Logistic Regression from Scratch Introduction to Scikit-learn in Python Train Logistic Regression in python Multiclass using Logistic Regression How to use Multinomial and Ordinal Logistic Regression in R ? Challenges with Linear Regression Introduction to Regularisation Implementing Regularisation Ridge Regression Lasso Regression
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naive Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
Understand Basics of Recommendation Engine with Case Study
8 Ways to Improve Accuracy of Machine Learning Models
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
Model Deployment in Android Model Deployment in Iphone

What are Logarithms and Exponents in Complexity Analysis?

Mounish V
Mounish V Last Updated : 01 Jul, 2024
4 min read

Introduction

Logarithms and exponents are crucial in evaluating the efficiency of algorithms in computer science. This article discusses these mathematical concepts, detailing their significance in complexity analysis and offering practical examples to demonstrate their applications. Let’s also see and understand how logarithms and exponents impact algorithm performance.

Complexity Analysis

Overview

  • Learn the basics of logarithms and exponents.
  • Understand the role of binary logarithms.
  • Learn how logarithms and exponents relate to complexity analysis.
  • Compare logarithmic and linear functions.
  • Apply these concepts in practical examples, such as binary search.

Table of contents

What are Logarithms and Exponents?

Logarithms and exponents are inverse operations. While exponents deal with repeated multiplication, logarithms find the exponent that produces a given number. These concepts are fundamental in computer science, particularly in analyzing algorithms’ efficiency.

Prerequisites

  • Exponent: The power to which a number (base) is raised.
  • Base: The number being multiplied by itself.
  • Common Logarithm: A logarithm with base 10.
  • Binary Logarithm: A logarithm with base 2, crucial in computer science.

Logarithms

A logarithm answers the question: To what power must a base number be raised to produce a given number? Mathematically, ( logb(n) = y ) means ( by = n ). For instance, ( log20(8000) = 3 ) because ( 203 = 8000).

Exponents

Exponents represent the repeated multiplication of a base number. For example, ( 23 = 2 times 2 times 2 = 8 ). In complexity analysis, exponents help describe algorithms’ growth rates.

Complexity Analysis

In algorithm analysis, we often encounter logarithmic and exponential terms. Understanding these helps us evaluate how an algorithm’s runtime scales with input size.

Logarithmic Complexity

Logarithmic time complexity, denoted as ( O(log n) ), indicates that the number of operations grows very slowly as the input size increases. This is highly efficient, as seen in binary search.

Exponential Complexity

Exponential time complexity, denoted as (O(2n) ), means the number of operations doubles with each additional input element, leading to rapid growth and inefficiency for large inputs.

Computer Science and Binary Logarithms

Binary logarithms, or base-2 logarithms, are prevalent in computer science because many algorithms, like binary search and merge sort, involve repeatedly dividing data in half. This division reflects a binary logarithm’s behavior.

Why Binary Logarithms?

Binary logarithms are commonly used because they fit the binary nature of computer operations and data structures. Algorithms that halve their input size at each step, such as binary search, exhibit logarithmic time complexity.

Comparing Logarithmic and Linear Functions

Logarithms and Exponents

On an asymptotic graph, a linear function ( O(n) ) increases steadily with input size, while a logarithmic function ( O(log n) ) rises quickly at first but then slows down significantly. This difference underscores why logarithmic algorithms are more efficient for large inputs.

Binary search is an efficient algorithm for finding an element in a sorted array. It works by repeatedly dividing the search interval in half:

  • Compare the target value to the middle element.
  • If the target equals the middle element, return the index.
  • If the target is less, repeat the search in the lower half.
  • If the target is greater, repeat the search in the upper half.

Binary search has a logarithmic time complexity of ( O(log n) ), meaning it can efficiently handle large inputs.

Binary Search Example

Consider a sorted array of 1,024 elements. To find a target value using binary search, you would:

  • Check the middle element.
  • If incorrect, eliminate half the array from consideration.
  • Repeat until the target is found.

This process requires at most  ( log2(1024) = 10 ) steps, demonstrating efficiency.

Conclusion

Understanding logarithms and exponents is crucial for grasping how efficiently algorithms work. Logarithmic time complexity, which is particularly efficient for handling large amounts of data, is essential in computer science. When you learn these concepts, you can thoroughly analyze algorithms and find ways to make them faster and more effective. 

Embark on your Data Science journey with our Introduction to Python course! Whether you’re new to coding or data science, this beginner-friendly course will empower you with essential Python skills. Elevate your career by enrolling now for free! Unlock the potential of Python and kickstart your data science endeavors. Don’t miss out – enroll today!

Frequently Asked Questions

Q1. What is a logarithm? 

Ans. A logarithm defines the exponent required for a base number to produce another specified number.

Q2. Why are binary logarithms significant in computer science? 

Ans. Binary logarithms hold importance because numerous algorithms hinge on halving data, aligning with the binary operations fundamental to computing.

Q3. How does logarithmic complexity compare with linear complexity? 

Ans. Logarithmic complexity expands far more gradually than linear complexity, rendering logarithmic algorithms notably efficient for handling substantial inputs.

Q4. What’s an example of an algorithm with logarithmic complexity? 

Ans. Binary search is a notable algorithm showcasing logarithmic time complexity. It efficiently pinpoints elements within a sorted array by iteratively halving the search interval.

AlgorithmalgorithmsBASEComputer ScienceefficiencyLineartimetime and space complexity
Mounish V
Mounish V

I'm a tech enthusiast, graduated from Vellore Institute of Technology. I'm working as a Data Science Trainee right now. I am very much interested in Deep Learning and Generative AI.

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