Logarithmic and Exponential Functions

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)
  2. Understand the definition and properties of e^x and ln x, including their relationship as inverse functions and their graphs (including the graph of y = e^kx for both positive and negative values of k)
  3. Use logarithms to solve equations and inequalities in which the unknown appears in indices (e.g. 2^x < 5, 3 × 2^(3x-1) < 5, 3^(x+1) = 4^(2x-1))
  4. Use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept (e.g. y = kx^n gives ln y = ln k + n ln x; y = k(a^x) gives ln y = ln k + x ln a)

1. What are Logarithms?

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The Relationship Between Logarithms and Indices

A logarithm is simply another way of writing an index (or power). The two forms are directly connected.

Index form: If a^b = c, then we can write this in logarithmic form as:

log_a(c) = b

This is read as "log to the base a of c equals b".

What does this mean? The logarithm tells us the power we need to raise the base to in order to get a certain number.

Example 1:

  • We know that 2^3 = 8
  • In logarithmic form: log₂(8) = 3
  • This means: "The power you raise 2 to in order to get 8 is 3"

Example 2:

  • We know that 10^2 = 100
  • In logarithmic form: log₁₀(100) = 2
  • This means: "The power you raise 10 to in order to get 100 is 2"

Example 3:

  • We know that 5^0 = 1
  • In logarithmic form: log₅(1) = 0
  • This means: "The power you raise 5 to in order to get 1 is 0"

Key Rule: For any positive base a (where a ≠ 1):

  • a^b = c is the same as log_a(c) = b

2. The Laws of Logarithms

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Just like we have rules for working with indices, we have rules for working with logarithms. These laws make calculations much easier.

Law 1: The Multiplication Law

log_a(x) + log_a(y) = log_a(xy)

When you add two logarithms with the same base, you can combine them by multiplying the numbers inside.

Example:

  • log₂(8) + log₂(4) = log₂(8 × 4) = log₂(32) = 5
  • Check: 2^5 = 32 ✓

Law 2: The Division Law

log_a(x) − log_a(y) = log_a(x/y)

When you subtract two logarithms with the same base, you can combine them by dividing the numbers inside.

Example:

  • log₃(27) − log₃(9) = log₃(27/9) = log₃(3) = 1
  • Check: 3^1 = 3 ✓

Law 3: The Power Law

log_a(x^n) = n log_a(x)

When you have a logarithm of a number raised to a power, you can bring the power out to the front as a multiplier.

Example:

  • log₂(16) = log₂(2^4) = 4 log₂(2) = 4 × 1 = 4
  • Check: 2^4 = 16 ✓

Special Values to Remember

For any base a (where a > 0 and a ≠ 1):

  1. log_a(1) = 0 (because a^0 = 1)
  2. log_a(a) = 1 (because a^1 = a)
  3. log_a(a^k) = k (this follows from the power law)

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