Unit 2 — Exponential & Logarithmic Functions
Unit 2 · Mathematics

The Beautiful World of
Exponents & Logarithms

Functions that grow, decay, and reverse — the mathematical language hiding inside nature, finance, and science.

y = aˣ
y = logₐx
e ≈ 2.718...
ln x = logₑx
01
Exponential Functions

Properties & Graphs

📐

Basic Form

When base a is positive and not equal to 1, a function of the form y = aˣ is called an exponential function.

a > 0, a ≠ 1
📈

When a > 1 — Growth

If the base is greater than 1, the function increases rapidly moving right. Classic examples: compound interest, population explosion.

e.g. y = 2ˣ, y = 10ˣ
📉

When 0 < a < 1 — Decay

If the base is between 0 and 1, the function decreases moving right. Used in radioactive decay and drug concentration models.

e.g. y = (1/2)ˣ
▸ Graph comparison — y = 2ˣ (blue) vs y = (½)ˣ (red)
(0, 1) y = 2ˣ y=(½)ˣ y = 1 0 1 2 −1
Domain: all real numbers (−∞, +∞)
Range: positive reals (0, +∞)
Asymptote: y = 0 (x-axis)
Always passes through (0, 1)
Worked Example 1
Solve for x: 23x−1 = 16
①
Rewrite the right side as a power of 2: 16 = 2⁴
②
Same base → set exponents equal: 3x − 1 = 4
③
Solve for x: 3x = 5
∴ x = 5/3
02
Logarithmic Functions

Properties & Graphs

y = loga x
This is the inverse function of y = aˣ — the x and y axes simply swap roles!
🔄

The Log–Exponent Connection

Exponential and logarithmic forms are freely interchangeable. This link is the foundation of everything!

aʸ = x ⟺ y = logₐx
📋

The Power of Log Rules

Logs turn multiplication into addition, division into subtraction, and powers into multiplication. Pure magic for complex calculations!

logₐ(mn) = logₐm + logₐn
🎯

Graph Characteristics

Always passes through (1, 0). The y-axis (x = 0) is a vertical asymptote. Only defined for x > 0.

Domain: x > 0
▸ Log function graphs — y = log₂x (blue) vs y = log½x (red)
(1, 0) y=log₂x y=log½x asymptote x=0 1 2 4 8
Worked Example 2
Evaluate: log₂ 48 − log₂ 3
①
Apply the quotient rule: log₂ 48 − log₂ 3 = log₂(48/3)
②
Simplify 48 ÷ 3 = 16: = log₂ 16
③
Since 16 = 2⁴: log₂ 2⁴ = 4
∴ Answer = 4
03
Natural Logarithm

The Natural Log ln and Euler's Number e

✨

Euler's Number e

A constant that emerges naturally from the limit of continuous compounding. One of the most important numbers in mathematics, alongside π!

e = lim(1 + 1/n)ⁿ ≈ 2.71828...
🌿

The Natural Log ln

A logarithm with base e is called the natural logarithm and written as ln. The most widely used log in science, engineering, and economics!

ln x = logₑ x
⚡ Watch (1 + 1/n)ⁿ converge to e as n grows larger
10
2.59374
Computed value of (1 + 1/n)ⁿ — gets closer to e = 2.71828... as n increases
💡
4 Essential Identities to Memorize — ln(eˣ) = x  ·  e^(ln x) = x  ·  ln 1 = 0  ·  ln e = 1 — These appear on virtually every exam!
PropertyExponential eˣNatural Log ln x
Domain(−∞, +∞)(0, +∞)
Range(0, +∞)(−∞, +∞)
Key valuee⁰ = 1ln 1 = 0
Asymptotey = 0x = 0
RelationshipInverse functions — symmetric about y = x
Worked Example 3
Evaluate: ln(e³) + ln(1/e)
①
First term: ln(e³) = 3  (since ln(eˣ) = x)
②
Second term: ln(1/e) = ln(e⁻¹) = −1
③
Add them: 3 + (−1) = 2
∴ Answer = 2
04
Solving Equations

Solving Exponential & Logarithmic Equations

🔑

Strategy 1 — Common Base

Rewrite both sides with the same base, then use the fact that equal bases mean equal exponents.

aᵐ = aⁿ → m = n
🪄

Strategy 2 — Take the Log

Apply log (or ln) to both sides to bring the exponent down. Especially powerful when bases differ.

Apply ln to both sides
♻️

Strategy 3 — Convert to Exponential

Rewrite a logarithmic equation in exponential form to solve for the unknown.

logₐx = b → x = aᵇ
Worked Example 4 — Exponential Equation
Solve 3ˣ = 20 to three decimal places.
①
Take ln of both sides: ln(3ˣ) = ln 20
②
Power rule brings exponent down: x · ln 3 = ln 20
③
Divide both sides by ln 3: x = ln 20 / ln 3
④
Calculate: x = 2.9957 / 1.0986 ≈ 2.727
∴ x ≈ 2.727
Worked Example 5 — Logarithmic Equation
Solve: log₃(x − 2) + log₃(x + 4) = 3
①
Combine using the product rule: log₃[(x−2)(x+4)] = 3
②
Convert to exponential form: (x−2)(x+4) = 3³ = 27
③
Expand: x² + 2x − 8 = 27 → x² + 2x − 35 = 0
④
Factor: (x+7)(x−5) = 0 → x = −7 or x = 5
⑤
Check domain condition (x − 2 > 0, so x > 2): x = −7 is rejected!
∴ x = 5
⚠️
Always check the domain!  The argument of a logarithm must always be positive (> 0). After solving, substitute your answer back and verify all arguments are positive — skipping this step costs marks!
05
Real-World Modeling

Population Growth, Half-Life & Compound Interest

🌍

Population Growth

When an initial population P₀ grows continuously at rate r, this formula gives the population after t years. The quintessential exponential application.

P(t) = P₀ · eʳᵗ
☢️

Half-Life (Radioactive Decay)

Given half-life T, this gives the remaining amount at time t. Used across medicine, archaeology, and nuclear physics.

A(t) = A₀ · (½)^(t/T)
💰

Continuous Compounding

Principal P invested at annual rate r for t years with continuous compounding. The ideal limit of compound interest naturally produces e!

A = P · eʳᵗ
💰 Continuous Compound Interest Calculator — drag the sliders
$10,000
7.0%
10 yrs
$20,138
Result using A = P · eʳᵗ
Worked Example 6 — Half-Life
A radioactive substance has a half-life of 5 years. How long does it take for 400g to decay to 50g?
①
Apply the half-life formula: 50 = 400 · (1/2)^(t/5)
②
Divide both sides by 400: (1/2)^(t/5) = 1/8 = (1/2)³
③
Same base → equal exponents: t/5 = 3
∴ t = 15 years
Unit 2 — Exponential & Logarithmic Functions  ·  All 5 topics complete ✓