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Machine Learning Basics for a Newbie
6 Steps of Machine learning LifecycleIntroduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data InsightsDescriptive StatisticsInferential StatisticsHow to Understand Population Distributions?
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Probability for Data ScienceBasic Concepts of ProbabilityAxioms of ProbabilityConditional Probability
Central Tendencies for Continuous VariablesSpread of DataKDE plots for Continuous VariableOverview of Distribution for Continuous variablesNormal DistributionSkewed DistributionSkeweness and KurtosisDistribution for Continuous Variable
Central Tendencies for Categorical VariablesUnderstanding Discrete DistributionsPerforming EDA on Categorical Variables
Dealing with Missing ValuesUnderstanding OutliersIdentifying Outliers in DataOutlier Detection in PythonOutliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and PopulationCentral Limit TheoremConfidence Interval and Margin of Error
Bivariate Analysis Introduction
CovariancePearson CorrelationSpearman's Correlation & Kendall's TauCorrelation versus CausationTabular and Graphical methods for Bivariate AnalysisPerforming Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical VariablesIntroduction to Hypothesis TestingP-valueTwo sample Z-testT-testT-test vs Z-testPerforming Bivariate Analysis on Continuous-Catagorical variables
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Multivariate AnalysisA Comprehensive Guide to Data ExplorationThe Data Science behind IPL
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Understanding Cost FunctionUnderstanding Gradient DescentMath Behind Gradient DescentAssumptions of Linear RegressionImplement Linear Regression from ScratchTrain Linear Regression in PythonImplementing Linear Regression in RDiagnosing Residual Plots in Linear Regression ModelsGeneralized Linear ModelsIntroduction to Logistic RegressionOdds RatioImplementing Logistic Regression from ScratchIntroduction to Scikit-learn in PythonTrain Logistic Regression in pythonMulticlass using Logistic RegressionHow to use Multinomial and Ordinal Logistic Regression in R ?Challenges with Linear RegressionIntroduction to RegularisationImplementing RegularisationRidge RegressionLasso Regression
Introduction to K Nearest NeighboursDetermining the Right Value of K in KNNImplement KNN from ScratchImplement KNN in Python
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Introduction to Feature SelectionFeature Selection AlgorithmsMissing Value RatioLow Variance FilterHigh Correlation FilterBackward Feature EliminationForward Feature SelectionImplement Feature Selection in PythonImplement Feature Selection in R
Introduction to Decision TreePurity in Decision TreeTerminologies Related to Decision TreeHow to Select Best Split Point in Decision Tree?Chi-SquaresInformation GainReduction in VarianceOptimizing Performance of Decision TreeTrain Decision Tree using Scikit LearnPruning of Decision Trees
Introduction to Feature EngineeringFeature TransformationFeature ScalingFeature EngineeringFrequency EncodingAutomated Feature Engineering: Feature Tools
Introduction to Naive BayesConditional Probability and Bayes TheoremIntroduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings!Working of Naive BayesMath behind Naive BayesTypes of Naive BayesImplementation of Naive Bayes
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Introduction to Ensemble TechniquesBasic Ensemble TechniquesImplementing Basic Ensemble TechniquesFinding Optimal Weights of Ensemble Learner using Neural NetworkWhy Ensemble Models Work well?
Introduction to StackingImplementing StackingVariants of StackingImplementing Variants of StackingIntroduction to BlendingBootstrap SamplingIntroduction to Random SamplingHyper-parameters of Random ForestImplementing Random ForestOut-of-Bag (OOB) Score in the Random ForestIPL Team Win Prediction Project Using Machine LearningIntroduction to BoostingGradient Boosting AlgorithmMath behind GBMImplementing GBM in pythonRegularized Greedy ForestsExtreme Gradient BoostingImplementing XGBM in pythonTuning Hyperparameters of XGBoost in PythonImplement XGBM in R/H2OAdaptive BoostingImplementing Adaptive BoosingLightGBMImplementing LightGBM in PythonCatboostImplementing Catboost in Python
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Introduction to Principal Component AnalysisSteps to Perform Principal Compound AnalysisComputation of Covariance MatrixFinding Eigenvectors and EigenvaluesImplementing PCA in pythonVisualizing PCAA Brief Introduction to Linear Discriminant AnalysisIntroduction to Factor Analysis
Introduction to ClusteringApplications of ClusteringEvaluation Metrics for ClusteringUnderstanding K-MeansImplementation of K-Means in PythonImplementation of K-Means in RChoosing Right Value for KProfiling Market Segments using K-Means ClusteringHierarchical ClusteringImplementation of Hierarchial ClusteringDBSCANDefining Similarity between clustersBuild Better and Accurate Clusters with Gaussian Mixture Models
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Introduction to Machine Learning InterpretabilityFramework and Interpretable Modelsmodel Agnostic Methods for InterpretabilityImplementing Interpretable ModelUnderstanding SHAPOut-of-Core MLIntroduction to Interpretable Machine Learning ModelsModel Agnostic Methods for InterpretabilityGame Theory & Shapley Values
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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.

Mounish V
Mounish V

Passionate about technology and innovation, a graduate of Vellore Institute of Technology. Currently working as a Data Science Trainee, focusing on Data Science. Deeply interested in Deep Learning and Generative AI, eager to explore cutting-edge techniques to solve complex problems and create impactful solutions.

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