# Logarithmic functions

> Edexcel International GCSE Further Pure Mathematics · 4PM1
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s1-logarithmic-functions/

This guide covers core logarithmic function content for Edexcel IGCSE Further Pure Math 4PM1: their definition, inverse relationship with exponential functions, and key graph sketching rules.

**Prerequisites:** Understanding of function inverses and basic function notation; Ability to plot and label basic function graphs

## Learning objectives

- Recognise exponential $a^x$ and logarithmic $\log_b x$ as mutually inverse functions
- Apply the definition $\log_b x = y \iff b^y = x$ to convert between exponential and log form
- Sketch graphs of $a^x$ and $\log_b x$, identifying key points and asymptotes

## Inverse Relationship & Logarithm Definition

Exponential functions of the form $y = a^x$ (where $a > 1$) and logarithmic functions of the form $y = \log_b x$ (where $b > 1$) are mutually inverse functions. This means one undoes the operation of the other, and their graphs are reflections of each other across the line $y = x$.

**Logarithm Definition** — Equivalent to $b^y = x$, where $b > 1$ is the base, $x > 0$ is the input, and $y$ is the resulting exponent.

*Notation:* \log_b x = y

*Example:* $\log_2 8 = 3$ because $2^3 = 8$

**Worked example:** Write the exponential statement $3^4 = 81$ in logarithmic form, and the logarithmic statement $\log_5 125 = 3$ in exponential form.

1. Recall the core definition: $\log_b x = y \iff b^y = x$. For $3^4 = 81$, identify $b=3$, $y=4$, $x=81$.
2. Substitute into the logarithmic form to get $\log_3 81 = 4$.
3. For $\log_5 125 = 3$, identify $b=5$, $y=3$, $x=125$.
4. Substitute into the exponential form to get $5^3 = 125$.

> **Exam tip:** Always check the input to a logarithm is positive, as logs of non-positive values are undefined for real numbers at this level.

*Calculator:* allowed

## Graphs of Exponential Functions $y = a^x$

All exponential functions with base $a > 1$ have a consistent recognisable shape. They pass through the point $(0,1)$ because any positive number raised to the power of 0 equals 1. The graph increases as $x$ increases, and approaches the horizontal asymptote $y = 0$ as $x$ becomes very negative.

**Worked example:** Sketch the graph of $y = 2^x$, marking the key point and asymptote clearly.

1. Mark the y-intercept at $(0,1)$, since $2^0 = 1$.
2. Draw the horizontal asymptote as a dashed line at $y = 0$ (the x-axis), as the graph never touches or crosses this line.
3. Draw a smooth increasing curve that gets closer to $y=0$ as $x \to -\infty$, and rises steeply as $x$ becomes positive.
4. Label the curve, intercept and asymptote clearly for full marks.

> **tip**
>
> The graph of $y = a^x$ is always in the upper half of the coordinate plane ($y > 0$) for all real values of $x$.

> **Exam tip:** You only need to draw the correct shape, label the key point and asymptote for graph questions unless explicitly asked to plot specific points.

*Calculator:* allowed

## Graphs of Logarithmic Functions $y = \log_b x$

As the inverse of exponential functions, log graphs are reflections of $y = b^x$ across the line $y=x$. All log graphs with base $b > 1$ pass through the point $(1,0)$ because $\log_b 1 = 0$ (since $b^0 = 1$). They have a vertical asymptote at $x = 0$ (the y-axis) as $x$ approaches 0 from the positive side, and increase slowly as $x$ grows.

**Worked example:** Sketch the graph of $y = \log_3 x$, marking the key point and asymptote, and state its relationship to $y = 3^x$.

1. Mark the x-intercept at $(1,0)$, since $\log_3 1 = 0$.
2. Draw the vertical asymptote as a dashed line at $x = 0$ (the y-axis), as the log function is only defined for positive $x$ values.
3. Draw a smooth increasing curve that gets closer to $x=0$ as $x \to 0^+$, and rises gradually as $x$ increases.
4. State that $y = \log_3 x$ is the reflection of $y = 3^x$ across the line $y = x$.

> **Exam tip:** Never draw your log graph entering the region $x < 0$, as logarithms of non-positive numbers are undefined at IGCSE level.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Writing $\log_b 0$ or $\log_b (-x)$ as valid values
  - Why it fails: Logarithms are only defined for positive inputs, as no positive base raised to a real power equals zero or a negative number
  - Correct: Always restrict the input to log functions to $x > 0$, and state this explicitly if asked for the domain
- **Wrong:** Drawing the asymptote of $y = \log_b x$ as $y=0$ instead of $x=0$
  - Why it fails: Confusing the horizontal asymptote of exponential functions with the vertical asymptote of their inverse log functions
  - Correct: Remember: exp graphs have horizontal asymptote $y=0$, log graphs have vertical asymptote $x=0$
- **Wrong:** Stating the intercept of $y = \log_b x$ is $(0,1)$ instead of $(1,0)$
  - Why it fails: Mixing up the intercepts of inverse functions, which swap x and y coordinates
  - Correct: The intercept of $y = a^x$ is $(0,1)$, so the inverse log function has intercept $(1,0)$
- **Wrong:** Converting $b^y = x$ to $\log_y b = x$
  - Why it fails: Misplacing the base, input and output values in the log definition
  - Correct: Use the rule: the base of the exponent is the base of the logarithm, so $\log_b x = y$
- **Wrong:** Drawing the log graph as decreasing for $b>1$
  - Why it fails: Confusing out-of-scope logs with base <1 with the required base >1 functions for 4PM1
  - Correct: All log functions tested in 4PM1 have base $b>1$, so they are strictly increasing

## Cheatsheet

| Function | Key Point | Asymptote | Shape (base >1) |
| --- | --- | --- | --- |
| $y = a^x$ | $(0, 1)$ | $y = 0$ (horizontal) | Increasing, steep rise for positive x |
| $y = \log_b x$ | $(1, 0)$ | $x = 0$ (vertical) | Increasing, slow rise for large x |
| Inverse Relationship | Swap x/y coordinates | - | Reflection across line $y=x$ |

## What's next

Now that you have mastered the foundational definition and graph properties of logarithmic functions, you are ready to move on to applying logarithm laws to simplify expressions and solve equations, the next sub-topic in the S1 unit. This knowledge is critical for all subsequent algebra and function topics in your 4PM1 exam, as logarithms are often used alongside indices and polynomials in multi-step questions. You will also encounter logarithmic relationships in later topics such as geometric sequences, where logs are used to solve for unknown terms or the number of terms in a sequence. Make sure you can reliably convert between exponential and log form and sketch both graphs quickly before moving on, as these skills are regularly tested in low-mark foundational questions at the start of exam papers.

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