1 Laws of Exponents
| Law | Formula | Example |
|---|---|---|
| Product | aᵐ · aⁿ = aᵐ⁺ⁿ | x³ · x⁴ = x⁷ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | x⁵ ÷ x² = x³ |
| Power of power | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Power of product | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
| Power of quotient | (a/b)ⁿ = aⁿ/bⁿ | (x/3)² = x²/9 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | x⁻³ = 1/x³ |
| Fractional exponent | a^(m/n) = ⁿ√(aᵐ) | 8^(2/3) = (∛8)² = 4 |
| Fractional exponent alt. | a^(1/n) = ⁿ√a | 27^(1/3) = ∛27 = 3 |
2 Exponential Functions
The exponential function has the form y = a · cˣ or equivalently y = a(1 + r)ˣ where c = 1 + r.
General form (MHS)
y = a · cˣ
a — initial value
y-intercept = (0, a)
c — base
c > 1 → growth | 0 < c < 1 → decay
r — rate
c = 1 + r (r > 0 = growth, r < 0 = decay)
Key Characteristics
| Feature | Value |
|---|---|
| Domain | All real numbers (−∞, +∞) |
| Range | y > 0 (function is always positive) |
| Y-intercept | (0, a) |
| Horizontal asymptote | y = 0 (the x-axis) |
| Shape | Increasing if c > 1; Decreasing if 0 < c < 1 |
3 Growth & Decay Applications
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Compound interest formula: y = a(1 + r)ᵗ
a = initial amount · r = annual rate · t = time in years
a = initial amount · r = annual rate · t = time in years
Growth (c > 1, r > 0)
- c > 1 means the amount grows over time.
- Example: y = 1000(1.05)ᵗ — investment at 5% annual interest.
- To find when y doubles: set y = 2a, solve for t using logarithms.
Decay (0 < c < 1, r < 0)
- 0 < c < 1 means the amount shrinks over time.
- Example: y = 500(0.88)ᵗ — radioactive decay, 12% per period.
- The function never reaches zero — it approaches the x-axis asymptotically.
Finding the Rule from a Graph
- Identify the y-intercept → gives a.
- Pick a second known point (x, y).
- Substitute into y = a · cˣ and solve for c.
4 Introduction to Logarithms
A logarithm is the inverse operation of exponentiation.
💡
Fundamental definition: logₐ(b) = c ⟺ aᶜ = b
"The log base a of b equals c" means "a to the power c equals b"
"The log base a of b equals c" means "a to the power c equals b"
Convert to exponential
log₂(8) = 3 ↔ 2³ = 8
Common log (base 10)
log(x) = log₁₀(x)
Natural log (base e)
ln(x) = logₑ(x)
Useful identities
logₐ(aˣ) = x and a^(logₐ x) = x
Change of base
logₐ(b) = log(b)/log(a) = ln(b)/ln(a)
5 Laws of Logarithms
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All three laws work because logarithms are exponents — they inherit the laws of exponents.
Product rule
logₐ(MN) = logₐ(M) + logₐ(N)
Quotient rule
logₐ(M/N) = logₐ(M) − logₐ(N)
Power rule
logₐ(Mⁿ) = n · logₐ(M)
Expanding vs Condensing
| Task | What to do | Example |
|---|---|---|
| Expand | Apply rules to break apart a single log | log(xy²) = log x + 2 log y |
| Condense | Apply rules in reverse to write as single log | log x + 2 log y = log(xy²) |
| Simplify | Evaluate if possible, otherwise expand/condense | log₂(32) = log₂(2⁵) = 5 |
6 Solving Exponential Equations
- Method 1 (same base): If you can write both sides with the same base, set exponents equal.
- Method 2 (take log): If bases cannot be matched, apply log to both sides, then use power rule to bring exponent down.
- Method 3 (substitution): If equation looks like a quadratic in the exponent, use substitution u = aˣ.
Same base example
2^(x+1) = 2^3 → x + 1 = 3 → x = 2
Log both sides
3ˣ = 20 → log(3ˣ) = log(20) → x·log(3) = log(20) → x = log(20)/log(3)
Investment doubling
2A = A(1.05)ᵗ → 2 = 1.05ᵗ → log2 = t·log(1.05) → t ≈ 14.2 years
7 Common Mistakes
| Mistake | Fix |
|---|---|
| log(a + b) ≠ log(a) + log(b) | Product rule: log(ab) = log(a) + log(b). The sum inside a log cannot be split. |
| log(a − b) ≠ log(a) − log(b) | Only the product rule applies, not differences inside the log. |
| Power rule: distribute the coefficient | 5·log₂(6x) = 5·log₂(6) + 5·log₂(x) — the 5 multiplies EVERY term. |
| Forgetting base when converting | log₂(8) = 3 means 2³ = 8. The base of the log is the base of the exponent. |
| Negative log doesn't mean negative answer | log(0.5) is negative but 0.5 is a valid input. Domain of log: x > 0. |