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CIE A-Level Maths Study Notes

2.2.4 Logarithmic Transformation and Linearization

A logarithmic transformation is a key technique used to linearise non-linear relationships. This method greatly simplifies equations, aiding in the identification of unknown constants and facilitating the analysis of equations in a linear form.

Essence of Logarithmic Transformation

  • Purpose: To convert non-linear equations into linear ones.
  • Method: Application of logarithms to both sides of the equation.
  • Benefit: Simplifies equations and assists in determining unknown constants through the analysis of the linear form's gradient and intercept.

Transforming Equations

Example 1: Transforming y=kxny = kx^n

1. Original Equation: y=kxny = kx^n

2. Logarithmic Application: Taking the natural logarithm (ln) of both sides.

3. Transformed Equation: ln(y)=ln(k)+nln(x)\ln(y) = \ln(k) + n \ln(x)

4. Analysis: The equation now resembles a linear form y=mx+cy = mx + c, where m=nm = n and c=ln(k)c = \ln(k).

Example 2: Transforming y=k(ax)y = k(a^x)

1. Original Equation: y=k(ax)y = k(a^x)

2. Applying Logarithms: Take the natural logarithm of both sides.

3. Transformed Equation: ln(y)=ln(k)+xln(a)\ln(y) = \ln(k) + x \ln(a)

4. Analysis: This equation is also linearized, with m=ln(a)m = \ln(a) and c=ln(k).c = \ln(k).

Practical Applications of Linearisation

  • Curve Fitting: Transforms non-linear data into a linear form for enhanced analysis and fitting.
  • Modelling Exponential Growth: Crucial for understanding and predicting growth patterns.
  • Interpreting Logarithmic Scales: Such as the Richter scale for earthquakes.

Examples

Example 1: Linearising y=3x2y = 3x^2

1. Original Equation: y=3x2y = 3x^2

2. Logarithmic Application: ln(y)=ln(3x2)\ln(y) = \ln(3x^2)

3. Using Logarithmic Properties: ln(y)=ln(3)+2ln(x)\ln(y) = \ln(3) + 2 \ln(x)

4. Linear Form: ln(y)=2ln(x)+ln(3)\ln(y) = 2 \ln(x) + \ln(3), with m=2m = 2 and c=ln(3)c = \ln(3).

graph of logarithmic transformation

Example 2: Linearizing y=25xy = 2 \cdot 5^x

1. Original Equation: y=25x y = 2 \cdot 5^x

2. Applying Logarithms: ln(y)=ln(25x)\ln(y) = \ln(2 \cdot 5^x)

3. Using Logarithmic Properties:

ln(y)=ln(2)+ln(5x)\ln(y) = \ln(2) + \ln(5^x)

ln(y)=ln(2)+xln(5)\ln(y) = \ln(2) + x \cdot \ln(5)

4. Linear Form: ln(y)=mx+c\ln(y) = m \cdot x + c , with m=ln(5)m = \ln(5) and c=ln(2)c = \ln(2).

logarithmic transformation graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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