Exponential and Logarithmic Functions
Chapter 4 — Precalculus
Exponential and logarithmic functions appear everywhere in math, science, finance, and biology. This guide covers the core concepts, log laws, solving techniques, and real-world applications.
Chapter 4 Practice Problems
50+ questions covering all 6 sections
Chapter 4 Topics
4.1 — Exponential Functions
Definition f(x) = bˣ, graphs, transformations, base e, domain/range
4.2 — Logarithmic Functions
Definition as inverse of exponential, graphs, domains, common log, natural log
4.3 — Laws of Logarithms
Product, quotient, power rules; change of base; expanding and condensing log expressions
4.4 — Exponential Equations
Solving bˣ = bʸ, solving with logs, using ln or log₁₀
4.5 — Logarithmic Equations
Solving log_b(x) = c, extraneous solutions, checking domain
4.6 — Modeling with Exponentials
Growth/decay, compound interest, half-life, Newton's cooling law, doubling time
Exponential Functions: The Key Rules
For f(x) = bx where b > 0 and b ≠ 1:
- • If b > 1: exponential growth — function increases, passes through (0, 1)
- • If 0 < b < 1: exponential decay — function decreases, passes through (0, 1)
- • Domain: all real numbers; Range: (0, ∞)
- • Horizontal asymptote at y = 0 (unless shifted)
- • The natural base e ≈ 2.71828 is most important in calculus
Common Mistake
Wrong: 2⁻³ = −8
Right: 2⁻³ = 1/2³ = 1/8 → Negative exponents mean reciprocal, not negative value
Logarithm Laws — Reference Card
Product rule
log_b(MN) = log_b(M) + log_b(N)
Log of a product = sum of logs
Quotient rule
log_b(M/N) = log_b(M) − log_b(N)
Log of a quotient = difference of logs
Power rule
log_b(Mⁿ) = n · log_b(M)
Bring the exponent down as a multiplier
Change of base
log_b(x) = log(x) / log(b)
Evaluate any log on a calculator using log₁₀ or ln
Identity
log_b(b) = 1
log base b of b always equals 1
Zero log
log_b(1) = 0
log base b of 1 always equals 0
Inverse property
b^(log_b(x)) = x
Exponential and log undo each other
How to Solve Exp/Log Equations
Solving Exponential Equations
Strategy 1: Same base
4ˣ = 64
4ˣ = 4³
x = 3
Strategy 2: Take log
3ˣ = 50
x·ln(3) = ln(50)
x = ln(50)/ln(3) ≈ 3.56
Strategy 3: e as base
eˣ = 20
ln(eˣ) = ln(20)
x = ln(20) ≈ 3.00
Solving Logarithmic Equations
Strategy 1: Exponentiate
log₂(x) = 5
2⁵ = x
x = 32
Strategy 2: Same base
log(x) = log(25)
x = 25
⚠ Check for extraneous solutions
After solving, check that all arguments of log are positive. log(x-2) requires x > 2.
Real-World Applications
These formulas are tested heavily on precalculus exams. Know which formula to use and what each variable means.
Compound Interest
A = P(1 + r/n)^(nt)
P = principal, r = annual rate, n = compounds/year, t = years
Example: $1,000 at 5% compounded monthly for 10 years: A = 1000(1 + 0.05/12)^120 ≈ $1,647.01
Continuous Compound Interest
A = Pe^(rt)
P = principal, r = annual rate, t = years, e ≈ 2.718
Example: $1,000 at 5% continuously for 10 years: A = 1000·e^0.5 ≈ $1,648.72
Exponential Growth/Decay
P(t) = P₀ · e^(kt)
P₀ = initial amount, k > 0 (growth), k < 0 (decay), t = time
Example: Population starts at 10,000, grows at 3%/year: P(t) = 10000·e^(0.03t)
Half-Life
A(t) = A₀ · (1/2)^(t/h)
A₀ = initial amount, h = half-life period, t = time
Example: Carbon-14 has half-life 5,730 years. After 11,460 years: A = A₀·(1/2)² = A₀/4
Practice Exponential & Log Problems
Chapter 4 in NailTheTest has 50+ practice problems covering all 6 sections — growth/decay, log laws, solving equations, and modeling. Step-by-step solutions and private tutoring included.