Edexcel IAL - Pure Maths 3- 3.2 The function ln x and its graph- Study notes  - New syllabus

The Natural Logarithmic Function \( \ln x \)

The function \( y = \ln x \) is called the natural logarithmic function.

It is defined as the inverse function of the exponential function \( y = e^x \).

\( \ln x \) as the Inverse of \( e^x \)

If:

\( y = e^x \)

then its inverse is:

\( y = \ln x \)

This means:

\( \ln(e^x) = x \quad \text{and} \quad e^{\ln x} = x \)

The graphs of \( y = e^x \) and \( y = \ln x \) are reflections of each other in the line:

\( y = x \)

Key Properties of \( y = \ln x \)

Graph of \( y = \ln x \)

• Passes through \( (1,0) \)
• Approaches the y-axis as \( x \to 0^+ \)
• Increases slowly for large \( x \)
• Defined only for positive \( x \)

Solving Exponential Equations

Equations of the form \( e^{ax+b} = p \)

Method:

• Take natural logarithms on both sides
• Use \( \ln(e^k) = k \)
• Solve for \( x \)

\( e^{ax+b} = p \Rightarrow ax + b = \ln p \)

Solving Logarithmic Equations

Equations of the form \( \ln(ax+b) = q \)

Method:

• Rewrite using the exponential form
• Solve for \( x \)
• Check the domain \( ax+b>0 \)

\( \ln(ax+b) = q \Rightarrow ax+b = e^q \)