Logarithmic functions

Q: Do I need to differentiate logarithmic functions for 4PM1?

No, differentiation of logarithmic or non- exponential functions is not part of the 4PM1 specification. You only need to know the shape of their graphs and their inverse relationship.

Edexcel International GCSE Further Pure Mathematics· 4PM1 Section S1 (2016 spec)· 12 min read

1. Inverse Relationship & Logarithm Definition★★☆☆☆⏱ 4 min

✓ Calculator OK

Exponential functions of the form $y = a^{x}$ (where $a > 1$ ) and logarithmic functions of the form $y = lo g_{b} x$ (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 $y = x$ .

📘 Definition

Logarithm Definition

\log_b x = y

Equivalent to $b^{y} = x$ , where $b > 1$ is the base, $x > 0$ is the input, and $y$ is the resulting exponent.

Example:

$lo g_{2} 8 = 3$ because $2^{3} = 8$

📐 Worked Example

Write the exponential statement $3^{4} = 81$ in logarithmic form, and the logarithmic statement $lo g_{5} 125 = 3$ in exponential form.

1
Recall the core definition: $lo g_{b} x = y ⟺ b^{y} = x$ . For $3^{4} = 81$ , identify $b = 3$ , $y = 4$ , $x = 81$ .
2
Substitute into the logarithmic form to get $lo g_{3} 81 = 4$ .
3
For $lo g_{5} 125 = 3$ , 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.

2. Graphs of Exponential Functions $y = a^x$★★★☆☆⏱ 3 min

✓ Calculator OK

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.

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.

3. Graphs of Logarithmic Functions $y = \log_b x$★★★☆☆⏱ 3 min

✓ Calculator OK

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 $lo g_{b} 1 = 0$ (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 $y = lo g_{3} x$ , marking the key point and asymptote, and state its relationship to $y = 3^{x}$ .

1
Mark the x-intercept at $(1, 0)$ , since $lo g_{3} 1 = 0$ .
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 $y = lo g_{3} x$ 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.

4. Common Pitfalls

Wrong move:

Writing $lo g_{b} 0$ or $lo g_{b} (- x)$ as valid values

Why:

Logarithms are only defined for positive inputs, as no positive base raised to a real power equals zero or a negative number

Correct move:

Always restrict the input to log functions to $x > 0$ , and state this explicitly if asked for the domain

Wrong move:

Drawing the asymptote of $y = lo g_{b} x$ as $y = 0$ instead of $x = 0$

Why:

Confusing the horizontal asymptote of exponential functions with the vertical asymptote of their inverse log functions

Correct move:

Remember: exp graphs have horizontal asymptote $y = 0$ , log graphs have vertical asymptote $x = 0$

Wrong move:

Stating the intercept of $y = lo g_{b} x$ is $(0, 1)$ instead of $(1, 0)$

Why:

Mixing up the intercepts of inverse functions, which swap x and y coordinates

Correct move:

The intercept of $y = a^{x}$ is $(0, 1)$ , so the inverse log function has intercept $(1, 0)$

Wrong move:

Converting $b^{y} = x$ to $lo g_{y} b = x$

Why:

Misplacing the base, input and output values in the log definition

Correct move:

Use the rule: the base of the exponent is the base of the logarithm, so $lo g_{b} x = y$

Wrong move:

Drawing the log graph as decreasing for $b > 1$

Why:

Confusing out-of-scope logs with base <1 with the required base >1 functions for 4PM1

Correct move:

All log functions tested in 4PM1 have base $b > 1$ , so they are strictly increasing

5. Quick Reference Cheatsheet

Function	Key Point	Asymptote	Shape (base >1)
$y = a^{x}$	$(0, 1)$	$y = 0$ (horizontal)	Increasing, steep rise for positive x
$y = lo g_{b} x$	$(1, 0)$	$x = 0$ (vertical)	Increasing, slow rise for large x
Inverse Relationship	Swap x/y coordinates		Reflection across line $y = x$

6. Frequently Asked

Do I need to differentiate logarithmic functions for 4PM1?

No, differentiation of logarithmic or non- $e^{k x}$ exponential functions is not part of the 4PM1 specification. You only need to know the shape of their graphs and their inverse relationship.

Is the change of base formula given in the 4PM1 exam?

Yes, the change of base formula for logarithms is provided on the 4PM1 formula sheet, though it is not required for this sub-topic (it is covered in S1_T02).

Going deeper

official_resourceEdexcel 4PM1 2016 SpecificationOfficial exam board specification for the 4PM1 qualification

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.