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 $y = kx^n$

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

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

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

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

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

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

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

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

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

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 $y = 3x^2$

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

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

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

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

Example 2: Linearizing $y = 2 \cdot 5^x$

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

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

3. Using Logarithmic Properties:

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

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

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