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Logarithmic & Exponential Functions

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Cambridge IGCSE (CIE) — Additional Maths

Logarithmic & Exponential Functions

The big idea: exponentials (like y = aˣ) describe things that grow or shrink by a constant multiplying factor — and logarithms are simply the "undo button" that lets you solve for the power when it's the unknown.

Summary — What's In This Chapter

Exponential functions — graphs of y = aˣ, growth vs decay, and why every one of them passes through (0,1).
The number e — the special base where the gradient of the curve equals the curve's own height.
Logarithmic functions — the inverse of exponentials; "the power you raise the base to, to get this number."
ln x — the natural logarithm, base e, and how it undoes eˣ.
Laws of logarithms — the addition, subtraction and power rules (mirrors of the laws of indices).
Change of base — how to evaluate logs with unusual bases, with or without a calculator.
Solving exponential equations — taking logs of both sides, including "hidden quadratic" traps.
Linearising relationships — turning curves like y = axⁿ and y = Abˣ into straight lines using logs.

1. Exponential Functions

An exponential function is one where the variable sits in the power (the exponent), not the base. It has the form:

General Form y = aˣ where a > 0

Read as: "a number a, raised to the power x."

Why every graph passes through (0, 1): Whatever base you pick, anything to the power 0 is 1. So a⁰ = 1 always — meaning y = aˣ always crosses the y-axis at the point (0, 1). This is one of the most testable facts in this chapter.

The x-axis is always an asymptote. As x gets very negative (for a > 1) or very positive (for 0 < a < 1), the curve gets closer and closer to y = 0 but never actually touches it. Think of it like a rubber band stretched toward the floor — it approaches but never lands.

Growth vs. Decay — the two shapes

Condition on a	Behaviour	Shape
a > 1	Exponential growth	Rises steeply to the right; flattens toward 0 on the left
0 < a < 1	Exponential decay	Falls steeply from the left; flattens toward 0 on the right
a = 1	Not considered	y = 1ˣ = 1 for all x — just a flat horizontal line, not really "exponential"

Visualising it:
For a > 1 (e.g. y = 2ˣ vs y = 3ˣ): the bigger base is the higher curve when x > 0, but the lower curve when x < 0.

For 0 < a < 1 (e.g. y = 0.2ˣ vs y = 0.3ˣ): it flips — the bigger base is lower when x > 0, and higher when x < 0.

Quick check

Notice that y = 0.25ˣ is a reflection in the y-axis of y = 4ˣ, because 0.25 = 1/4 = 4⁻¹. Spotting these reciprocal relationships saves a lot of sketching time in exams.

Practice Question 1

Sketch y = 5ˣ and y = 0.2ˣ on the same diagram. State one geometric relationship between the two curves, and mark the coordinates of any axis intercepts.

Practice Question 2

Without a calculator, explain why the graph of y = aˣ can never touch the x-axis, no matter what positive value a takes.

2. The Number "e" — A Very Special Base

Among all possible bases for an exponential function, one number is mathematically "perfect": e ≈ 2.718... It's an irrational number (like π), and it shows up naturally throughout maths, science, and finance.

What makes e so special? For the function y = eˣ, the gradient of the curve at any point is exactly equal to the y-value at that point. No other base has this clean property.

The defining property of e If y = eˣ, then dy/dx = eˣ

In plain English: the slope of the curve at any x-value is the same number as the height of the curve there. At x = 1, the curve's height is e ≈ 2.718, and its steepness at that exact point is also 2.718.

x	y = eˣ	dy/dx
−1	0.3678...	0.3678...
0	1	1
1	2.7182...	2.7182...
2	7.3890...	7.3890...

The negative exponential: y = e⁻ˣ is simply the reflection of y = eˣ in the y-axis — same idea as the 4ˣ / 0.25ˣ pair above.

Growth and Decay Models with e

Exponential growth y = Aeᵏˣ (k > 0) Exponential decay y = Ae⁻ᵏˣ (k > 0)

A = the initial (starting) value, at x = 0. k = a positive constant controlling how fast the growth or decay happens. The sign in front of kx tells you which one you've got — positive for growth, negative for decay.

Real-world connection

This is exactly the model used for population growth, radioactive decay, compound interest, and cooling objects — anywhere something changes at a rate proportional to its current size.

Practice Question 3

On the same diagram, sketch y = eˣ and y = e³ˣ. Which is steeper, and why? What do both curves have in common?

3. Logarithmic Functions

A logarithm is simply the inverse of raising a number to a power. If exponentials answer "what do I get if I raise a to the power x?", logarithms answer the reverse question: "what power do I need to raise a to, in order to get this number?"

Definition If a = bˣ, then log_b a = x (where a > 0, and b is called the base)

These two statements are completely equivalent — they say the exact same thing in two different languages.

How to "read" a logarithm out loud — this trick makes logs click for most students:

Say it like this

log₅(125) = 3 is read as: "The power you raise 5 to, to get 125, is 3." Check: 5³ = 125. ✓

Because logs and powers are inverses of each other, they "cancel out" when applied one after the other:

Cancellation identities log_a(aˣ) = x and a^{log_a x} = x

This is the same idea as √(x²) = x for square roots — doing an operation and then its exact inverse just gets you back where you started.

Why bother with logs? Equations like 2ˣ = 8 can be solved by inspection (x = 3, easy). But 2ˣ = 10 has no "nice" answer — there's no whole number or simple fraction that works. Logarithms let us express and calculate that exact answer: x = log₂10 ≈ 3.32.

Calculator Buttons You'll Actually Use

Button	What it computes
`log` or `log₁₀`	Logarithm base 10 (common logarithm)
`ln`	Natural logarithm (base e)
`log▢`	Logarithm with any base you type in

Watch out

(log x)² is NOT the same as log x². The first squares the whole log value; the second is log of x-squared, which by the power law equals 2 log x. Mixing these up is one of the most common exam errors.

Practice Question 4

Without using a calculator, evaluate log₂(64√2).

Practice Question 5

Use a calculator to evaluate log₁₀32 + log₂12 − log₃19, giving your answer to 3 significant figures.

4. ln x — The Natural Logarithm

ln stands for natural logarithm — it's just a logarithm where the base is e instead of 10 or anything else. It's so common it gets its own button and its own notation.

Definition ln x ≡ log_e x

Important: "ln" is a function (an instruction to do something to x), not a number by itself. Just like sin or cos, it needs an input.

Because ln x and eˣ are inverse functions of one another, they undo each other perfectly:

Key properties ln 1 = 0 | ln e = 1 | ln(eˣ) = x | e^{ln x} = x

ln x is only defined for x > 0 — you can never take the natural log (or any log) of zero or a negative number.

Solving Equations with e and ln

The golden rule: if your equation involves e, apply ln to both sides to bring the power down. If your equation involves ln, apply e to both sides (i.e., raise e to the power of each side) to remove it.

If you have...	Then...
eˣ = a	x = ln a
ln x = a	x = eᵃ
e^f(x) = g(x)	f(x) = ln g(x)
ln f(x) = g(x)	f(x) = e^g(x)

Exam strategy

On the non-calculator paper, you'll often be asked for an "exact value." That's your cue to leave the answer in terms of ln — don't try to force a decimal. e.g. x = (ln 5)/2 is a perfectly acceptable final answer.

Worked-style Practice Question 6

Solve the equation e²ˣ = 5, leaving your answer as an exact value.

Practice Question 7

Solve ln(2x − 1) = 3, giving your answer to 3 significant figures.

5. Laws of Logarithms

These laws let you simplify, combine, or split logarithmic expressions. They're not arbitrary — each one is a direct mirror of a corresponding law of indices. If you know your index laws, these should feel very familiar.

The Three Core Laws (for a, x, y > 0) log_a(xy) = log_ax + log_ay   →   mirrors aˣ × aʸ = aˣ⁺ʸ

log_a(x/y) = log_ax − log_ay   →   mirrors aˣ ÷ aʸ = aˣ⁻ʸ

log_a(xᵐ) = m·log_ax   →   mirrors (aˣ)ʸ = aˣʸ

Multiplying inside a log becomes adding outside it. Dividing inside becomes subtracting outside. A power inside a log can be pulled out the front as a multiplier.

These lead to a few extra results worth knowing cold:

Extra useful identities log_aa = 1 | log_a(aˣ) = x | a^log_ax = x | log_a1 = 0 | log_a(1/x) = −log_ax

The #1 trap in this topic

log_a(x + y) ≠ log_ax + log_ay. There is no law for logs of a — only for products, quotients, and powers. Students lose easy marks by "distributing" a log across a plus sign, which simply isn't allowed.

All of these laws apply equally to ln (since ln is just log base e) — including two especially handy shortcuts:

ln(eˣ) = x and e^{ln x} = x

Worked-style Practice Question 8

Write 3log₂(2x+3) + log₂5 − 2log₂(x+1) as a single logarithm.

Practice Question 9

Write the expression 2log 4 − log 2 in the form log k, where k is an integer. Hence solve 2log 4 − log 2 = −log(1/x).

6. Change of Base

The laws of logarithms only work when every log in your expression shares the same base. When bases don't match, you need to convert them first — that's what the change of base formula does.

Change of Base Formula log_ax = (log_bx) / (log_ba)

You can pick any base b you like — it always works. Choose it strategically: if a and x are both powers of some common number, pick b to be that number, and you can evaluate the whole thing without a calculator.

Example of smart base choice: To find log₈32 without a calculator, notice 8 = 2³ and 32 = 2⁵. Choosing b = 2:

log₈32 = (log₂32)/(log₂8) = 5/3 = 1⅔

This formula also gives you a useful "flip" identity, using the fact that log_xx = 1:

Reciprocal identity log_ax = 1 / log_xa

When you'll actually use this

Change of base is rare on a calculator paper (your calculator's log▢ button does it for you), but it's essential on the non-calculator paper, and whenever two logs in the same equation have different bases (e.g. combining log₃k and log₉n).

Practice Question 10

By choosing a suitable value for b, use the change of base law to find log₈32 without a calculator. (Try it yourself before checking!)

7. Solving Exponential Equations

An exponential equation has the unknown sitting in the power. Some can be solved by "spotting" a common base (e.g. 5²ˣ = 125 = 5³, so 2x = 3). Most, though, need logarithms.

The 4-Step Method STEP 1: Take logarithms of both sides
STEP 2: Use the laws of logarithms to bring the powers down
STEP 3: Rearrange to isolate x
STEP 4: Solve for x (calculate or leave exact)

Watch for "Hidden Quadratics"

Some exponential equations are secretly quadratics in disguise. The giveaway is a term like 4ˣ next to a term like 2ˣ — because 4ˣ = (2²)ˣ = (2ˣ)². If you substitute u = 2ˣ, the whole thing turns into an ordinary quadratic in u.

Spot these patterns 4ˣ = (2²)ˣ = 2²ˣ = (2ˣ)² | e²ˣ = (e²)ˣ = (eˣ)²

Worked example walkthrough

Solve 4ˣ − 3(2ˣ⁺¹) + 9 = 0.
1. Rewrite 4ˣ as (2ˣ)² and 2ˣ⁺¹ as 2(2ˣ).
2. Equation becomes (2ˣ)² − 6(2ˣ) + 9 = 0.
3. Substitute u = 2ˣ: u² − 6u + 9 = 0 → (u−3)² = 0 → u = 3.
4. So 2ˣ = 3. Take ln of both sides: x ln 2 = ln 3 → x = ln3/ln2 ≈ 1.58 (3 s.f.)

Practice Question 11

Solve 27ˣ = 9 without using a calculator, using the change of base approach.

Practice Question 12

Solve 9ˣ − 4(3ˣ) + 3 = 0.

8. Transforming Relationships to Linear Form

This is one of the most powerful applications of logarithms: turning a curved relationship into a straight line, so that experimental data (like giraffe sleeping habits, or radioactive decay measurements) can be analysed using simple straight-line techniques — finding a gradient and an intercept.

Case A: y = axⁿ (a power relationship)

Take ln of both sides and use the laws of logs to unpack it:

Linearising y = axⁿ ln y = ln a + n·ln x

Compare this to Y = mX + c. If you plot ln y (vertical axis) against ln x (horizontal axis), you get a straight line where the gradient m = n, and the y-intercept c = ln a.

Straight-line variable	Equals
Y	ln y
X	ln x
gradient m	n
intercept c	ln a → solve a = e^c

Case B: y = Abˣ (an exponential relationship)

Same idea, but this time take log (any base — log₁₀ is common) of both sides:

Linearising y = Abˣ log y = log A + x·log b

Plotting log y against x (not log x this time!) gives a straight line where the gradient m = log b, and the intercept c = log A.

Straight-line variable	Equals
Y	log y
X	x
gradient m	log b → solve b = 10^m
intercept c	log A → solve A = 10^c

How to tell the two cases apart

Ask: is the unknown constant in the base or the exponent? In y = axⁿ, the variable x is the base, and n is fixed — so both x and y get logged (ln x vs ln y). In y = Abˣ, x is the exponent — so only y gets logged, and x stays as-is on the horizontal axis.

Worked-style Practice Question 13

The graph of ln t against ln h is a straight line through (1, −0.9) and (4, −4.5), where t = ahᵇ. Find a and b in exact form.

Practice Question 14

When lg y is plotted against x, a straight line passes through (2, 5) and (5, 8). Show that y = A × bˣ, finding A and b.

What to Memorise

a⁰ = 1

Every exponential graph y = aˣ passes through (0, 1).

If a = bˣ then log_ba = x

The core definition connecting exponentials and logarithms.

ln x ≡ log_ex

Natural log, base e ≈ 2.718. ln(eˣ)=x and e^{ln x}=x.

log_a(xy) = log_ax + log_ay

Product law — multiplying inside becomes adding outside.

log_a(x/y) = log_ax − log_ay

Quotient law — dividing inside becomes subtracting outside.

log_a(xᵐ) = m log_ax

Power law — bring the exponent out the front.

log_ax = log_bx / log_ba

Change of base formula — pick b to make the numbers nice.

log_a(x+y) ≠ log_ax + log_ay

There is NO law for logs of a sum. This is the classic trap.

y = Aeᵏˣ / y = Ae⁻ᵏˣ

Growth (positive k) / decay (negative sign) models, A = initial value.

ln y = ln a + n ln x

Linear form of y = axⁿ; plot ln y vs ln x.

log y = log A + x log b

Linear form of y = Abˣ; plot log y vs x.

4ˣ = (2ˣ)², e²ˣ = (eˣ)²

Hidden quadratic patterns — substitute u to solve.

Concepts Checklist

I can sketch y = aˣ for a > 1 and 0 < a < 1, marking the intercept and asymptote.
I understand why every y = aˣ graph passes through (0, 1) and never touches the x-axis.
I know that y = eˣ has the unique property dy/dx = eˣ.
I can write growth/decay models in the form y = Aeᵏˣ or y = Ae⁻ᵏˣ and identify A and k.
I can convert between exponential form (a = bˣ) and log form (log_ba = x) fluently.
I can "read" a logarithm statement in plain English.
I know ln x is only defined for x > 0, and can use ln(eˣ)=x and e^{ln x}=x.
I can apply the product, quotient, and power laws of logarithms confidently.
I know that log_a(x+y) ≠ log_ax + log_ay — there is no "sum law."
I can use the change of base formula, choosing a smart base for non-calculator questions.
I can solve exponential equations by taking logs of both sides and rearranging.
I can spot "hidden quadratics" (like 4ˣ vs 2ˣ) and solve using substitution.
I can linearise y = axⁿ using ln y = ln a + n ln x and find a, n from a graph.
I can linearise y = Abˣ using log y = log A + x log b and find A, b from a graph.
I know when to leave an answer as an exact value (in terms of ln/e) vs a decimal.

Exam Tips & Common Mistakes

Don't split logs of sums. log(x + y) is NOT log x + log y. Students often try to "distribute" the log across a plus sign — there's simply no law for this. Only products, quotients, and powers can be split.

Don't confuse (log x)² with log x². These look similar but mean very different things — the first squares the output of the log, the second is the log of x squared (which equals 2 log x).

Always check your solutions are valid. Since log(x+k) is only defined when x > −k, some algebraic solutions to log equations turn out to be extraneous (invalid). Substitute back in and reject any that make you take the log of a negative number or zero.

Watch for "exact value" instructions. On non-calculator papers, don't try to evaluate ln 5 or e^0.3 as decimals — leave the answer exactly as it is. Converting to a decimal when an exact answer is requested loses marks.

Only use change of base when bases genuinely differ. It's a common over-correction to apply the change of base formula everywhere. Only reach for it when your logs don't already share a common base.

Match the linearising method to where the unknown sits. If the variable is in the base (y = axⁿ), log BOTH x and y. If the variable is in the exponent (y = Abˣ), log ONLY y and keep x as-is. Mixing these two up is a very common exam slip.

Look out for hidden quadratics before panicking. If you see terms like 9ˣ next to 3ˣ, or e²ˣ next to eˣ, recognise the squared relationship (9ˣ = (3ˣ)²) and use substitution (u = 3ˣ) to reduce it to a normal quadratic you already know how to solve.

Be familiar with your calculator's log buttons before the exam. Know the difference between the base-10 log button, the ln button, and the "any base" log▢ button — fumbling with this under time pressure costs valuable minutes.

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