Forgotten Graphs #1: ln(x)

Welcome to this series of blogs where I look at common graphs across the A level course which students struggle to recall during practice papers and even in the exam.

Today's blog focusses on ln(x).

The ln(x) graph can be drawn by effectively reflecting the y=e^x graph in the line y=x. In this respect ln(x) is the inverse function of the exponential function.

Graph showing relationship between e^x (blue), y=x (green) and ln(x) in red.

Most students are OK with recalling the exponential function but often do not remember the ln(x). I believe its down to a variety of factors:

• the ln(x) graph is not covered to a lot of detail in the Year 12 curriculum.

• there are not many questions in past papers where they ask students to transform the graph- this create a lack of exposure and hence lack of familiarity.

Lets focus on the important aspects of the ln(x) in more detail:

Graph of ln(x)

The key features of the ln(x) graph are:

• It cuts the x-axis at 1.

• It does not touch the y-axis, hence the y-axis is an asymptote.

• It increases in value in the positive y -axis after x=1.

• It has no horizontal asymptote.

Questions involving ln(x) don't come up regularly but when they do it can be linked with problems involving:

1. Transforming the ln(x) graph and finding the image of certain points after transformation.

2. Sketching the ln(x) as it is or after transformation.

3. Domain and Range.

4. Applying a modulus function on ln(x).

5. Iteration to find intercepts and points of intersection.

6. Testing your knowledge on the relationship of ln(x) and e^x.

Therefore its in your best interest to memorise this graph and its important features!