6.1 Log and Exponential Graphs / Properties

2026 Syllabus Objectives

By the end of these notes, you should be able to:

  • Know and use the basic properties and graphs of logarithmic and exponential functions, including ln x and e^x
  • Understand that f(x) = e^x and g(x) = ln x are inverses of each other
  • Understand what an asymptote is and how it applies to these graphs
  • State the equation of any asymptotes for these graphs
  • Sketch and interpret graphs of the form y = ke^(nx) + a and y = k ln(ax + b), where n, k, a, and b are integers

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1. What is an Exponential Function?

An exponential function is a function where the variable (like x) is the power (also called the exponent), not the base.

The general form looks like:

y = aˣ, where a is a positive number called the base

For example:

  • y = 2ˣ
  • y = 10ˣ
  • y = eˣ

The most important exponential function in mathematics is y = eˣ, where e is a special number approximately equal to 2.718. This number e appears naturally in many real-world situations (like population growth, radioactive decay, and compound interest).

2. The Graph of y = eˣ

Here are the key features of the graph of y = eˣ:

FeatureDetail
ShapeA smooth curve that rises steeply to the right
Passes through(0, 1) — because e⁰ = 1
As x → +∞y → +∞ (the graph rises without limit)
As x → −∞y → 0 (the graph gets closer and closer to zero, but never reaches it)
Asymptotey = 0 (the x-axis)
Always positivey > 0 for all values of x

Key idea: The graph of y = eˣ never crosses the x-axis. It just gets closer and closer to it as x becomes very negative. This is called asymptotic behaviour.

3. What is a Logarithmic Function?

A logarithm is the inverse of a power. It answers the question:

"What power do I need to raise the base to, in order to get this number?"

For example:

  • log₂(8) = 3, because 2³ = 8
  • log₁₀(100) = 2, because 10² = 100

Logarithms can be written to any base. The two most common bases you will use are:

  • log₁₀ — called the common logarithm (often just written as "log")
  • logₑ — called the natural logarithm, written as ln

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