📈

Understanding Exponentials and Logarithms

Apr 23, 2025

Year 1 Mathematics: Chapter 14 - Exponentials and Logarithms

Overview

  • Exponentials and logarithms are important and continue to be relevant in Year 2.
  • Focuses on understanding graph transformations, exponential growth/decay, and solving equations with exponentials and logarithms.

Exponential Graphs

  • Example: Exponential Growth: ( y = 2^x )
  • Exponential Decay: ( y = (\frac{1}{2})^x ) or ( y = 2^{-x} )
  • Graph Transformations:
    • Negative on outside: ( y = -2^x )
    • Negative exponent: ( y = -\frac{1}{2}^x )
  • Logarithmic Graph: ( y = \ln{x} ) or ( y = \log{x} )
    • Inverse of exponential graphs, reflected over line ( y = x ).

Differentiating Exponentials

  • Natural Exponent: ( e^x ) differentiates to ( e^x ).
  • General Form: ( e^{kx} ) differentiates to ( ke^{kx} ).
  • Other Bases: If base is ( a ), ( a^x ) differentiates to ( \ln{a} \times a^x ).

Logarithm Laws

  • Addition: ( \log{a} + \log{b} = \log{(ab)} )
  • Subtraction: ( \log{a} - \log{b} = \log{\frac{a}{b}} )
  • Power Rule: ( \log{a^b} = b\log{a} )
  • Cannot input negative numbers into a logarithm.

Solving Exponential and Logarithmic Equations

  • Convert to Logarithms:
    • ( 4^x = 7 ) becomes ( x = \log_4{7} ) or ( x = \frac{\ln{7}}{\ln{4}} ).
  • Logarithmic Equations:
    • Combine terms using log rules.
    • Convert log equation to exponential form.
  • Pseudo Quadratics: Manipulate exponential equations to form quadratics.

Solving Example Problems

  1. Example: Given ( 5^{x+1} = 5^{2x-6} ), solve by recognizing pseudo quadratics.
  2. Differentiation Example: Differentiate ( y = 4e^{-2x} ) to get ( \frac{dy}{dx} = -8e^{-2x} ).

Modeling with Exponentials

  • Use exponential models like ( P = ab^t ) or ( P = ae^{kx} ).
  • Example: Model population growth over time, solve for specific conditions.
  • Rate of Change: Differentiate to find change rate.
  • Limit: As ( t \to \infty ), exponential decay affects population reaching a limit.

Logarithmic and Non-Linear Data on Graphs

  • Convert exponential/non-linear data to linear using logs.
  • Steps:
    1. Take logarithms
    2. Find equation of the line
    3. Compare equations
    4. Solve for unknowns
  • Example: Given ( V = ab^t ), solve for A and B using linear graph info.

Additional Notes

  • Emphasis on practice, especially on exam-style questions.
  • Understanding graph shapes and transformations is crucial.
  • Encourage further study and practice to grasp depth of the topic.
  • Videos and resources are supplemental; core understanding is key.

Conclusion: Mastery of exponentials and logarithms in Year 1 sets a foundation for Year 2 topics. Practice and familiarity with transformations, solving equations, and applying laws are essential for success. Subscribe to further educational content for continued learning.

Full transcript