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 $c = \ln(2)$.

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.