Exponentials and Logarithms

What is the relationship between exponentials and logarithms?

A logarithm is the inverse of an exponential: if y = aˣ then x = logₐ y. The key laws are log a + log b = log ab, log a − log b = log(a/b) and n log a = log aⁿ. The natural logarithm ln x is the inverse of eˣ, and logs are used to solve equations such as 3ˣ = 20 and to linearise data.

Exponentials and Logarithms

Exponential and logarithmic functions are fundamental to A-Level Mathematics. They model growth and decay in real-world contexts and are essential tools for solving equations where the unknown is in the exponent.

Exponential Functions

An exponential function has the form y = aˣ where a > 0 and a ≠ 1. Key properties:

• The graph passes through (0, 1) since a⁰ = 1 for any valid base
• If a > 1, the function is increasing (exponential growth)
• If 0 < a < 1, the function is decreasing (exponential decay)
• The x-axis is a horizontal asymptote — the curve approaches but never touches y = 0
• The function is always positive: aˣ > 0 for all x

The most important exponential function uses the base e ≈ 2.71828, known as Euler's number. The function y = eˣ has the unique property that its derivative equals itself: d/dx(eˣ) = eˣ.

Logarithms

A logarithm is the inverse of an exponential. If aˣ = b, then logₐb = x (read "log base a of b equals x").

The natural logarithm ln(x) = logₑ(x) is the inverse of eˣ.

Laws of Logarithms

These laws hold for all valid bases:

Law	Statement
Product law	log(ab) = log(a) + log(b)
Quotient law	log(a/b) = log(a) - log(b)
Power law	log(aⁿ) = n·log(a)
Change of base	logₐb = logᵧb / logᵧa

Additionally: logₐa = 1, logₐ1 = 0, and logₐ(1/a) = -1.

Worked Example: Solve 3²ˣ⁺¹ = 5ˣ.

Step 1: Take natural logarithms of both sides: ln(3²ˣ⁺¹) = ln(5ˣ)

Step 2: Apply the power law: (2x + 1)ln3 = xln5

Step 3: Expand: 2xln3 + ln3 = xln5

Step 4: Collect x terms: 2xln3 - xln5 = -ln3

Step 5: Factorise: x(2ln3 - ln5) = -ln3

Step 6: Solve: x = -ln3 / (2ln3 - ln5) = -ln3 / (ln9 - ln5) = -ln3 / ln(9/5)

x ≈ -1.869

The Exponential Function and Natural Log

Key relationships between eˣ and ln(x):

• e^(ln x) = x for x > 0
• ln(eˣ) = x for all x
• The graph of y = ln(x) is a reflection of y = eˣ in the line y = x
• ln(x) has domain x > 0 and range all real numbers
• The y-axis is a vertical asymptote of ln(x)

Differentiation and Integration

Function	Derivative	Integral
eˣ	eˣ	eˣ + c
e^(kx)	ke^(kx)	(1/k)e^(kx) + c
ln(x)	1/x	x·ln(x) - x + c
aˣ	aˣ·ln(a)	aˣ/ln(a) + c

The result that the integral of 1/x is ln|x| + c is particularly important — it fills the gap in the power rule (which fails when n = -1).

Exponential Modelling

Many real-world quantities follow exponential models:

Growth: P = P₀·eᵏᵗ where k > 0 (population growth, compound interest, bacterial growth)

Decay: N = N₀·e⁻ᵏᵗ where k > 0 (radioactive decay, cooling, depreciation)

Worked Example: A radioactive substance has a half-life of 5 years. If the initial mass is 200g, find the mass after 12 years.

Step 1: Use N = N₀·e⁻ᵏᵗ. At t = 5, N = 100 (half of 200).

Step 2: 100 = 200·e⁻⁵ᵏ, so e⁻⁵ᵏ = 0.5

Step 3: -5k = ln(0.5), so k = -ln(0.5)/5 = ln2/5 ≈ 0.1386

Step 4: At t = 12: N = 200·e⁻⁰·¹³⁸⁶ˣ¹² = 200·e⁻¹·⁶⁶³ = 200 × 0.1895 ≈ 37.9g

Logarithmic Graphs for Modelling

If data follows y = axⁿ, then taking logs gives log(y) = log(a) + n·log(x). Plotting log(y) against log(x) gives a straight line with gradient n and y-intercept log(a).

If data follows y = abˣ, then log(y) = log(a) + x·log(b). Plotting log(y) against x gives a straight line with gradient log(b) and y-intercept log(a).

This technique lets you identify the model type and find constants from experimental data.

Exam Tips

• When solving exponential equations, always take logs — do not try to guess or use trial and improvement.
• Remember that ln and e are inverses. If you have e^(something) = value, apply ln to both sides.
• In modelling questions, clearly state the model and define your variables. Show the substitution to find constants.
• Watch for the domain of ln: you cannot take the log of zero or a negative number.
• When sketching, mark the key features: asymptotes, intercepts, and the relationship between eˣ and lnx as reflections.