Exponents & Logarithms

Cheat Sheet · Grade 11 Math
Emil Oliversen
LAWS OF EXPONENTS
Product
aᵐ · aⁿ = aᵐ⁺ⁿ
Quotient
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of power
(aᵐ)ⁿ = aᵐⁿ
Power of product
(ab)ⁿ = aⁿbⁿ
Zero exponent
a⁰ = 1
Negative exp
a⁻ⁿ = 1/aⁿ
Fraction exp
a^(m/n) = ⁿ√(aᵐ)
Root
a^(1/n) = ⁿ√a
EXPONENTIAL FUNCTION
General form
y = a · cˣ
a
y-intercept = (0, a)
c > 1
GROWTH
0 < c < 1
DECAY
Domain
all real numbers
Range
y > 0
Asymptote
y = 0 (x-axis)
LOGARITHM DEFINITION
logₐ(b) = c ⟺ aᶜ = b
Common log
log(x) = log₁₀(x)
Natural log
ln(x) = logₑ(x)
Change of base
logₐ(b) = log(b)/log(a)
Inverse
logₐ(aˣ) = x and a^(logₐ x) = x
LAWS OF LOGARITHMS
Product
logₐ(MN) = logₐM + logₐN
Quotient
logₐ(M/N) = logₐM − logₐN
Power
logₐ(Mⁿ) = n·logₐ(M)
WARNING: log(a+b) ≠ log(a)+log(b)
APPLICATIONS
Compound interest
y = a(1+r)ᵗ
a
initial amount
r
annual rate (as decimal)
t
time in years
SOLVING EXPONENTIAL EQ
  • Same base: equate exponents directly
  • Different base: take log of both sides
  • Use power rule to bring exponent down
  • Isolate and solve for variable
Example: 3ˣ=20
log(3ˣ)=log(20)
x·log(3)=log(20)
x = log(20)/log(3) ≈ 2.73
EXPAND vs CONDENSE
Expand
Break apart using the 3 laws
Condense
Reverse → write as single log
COMMON MISTAKES
  • log(a+b) ≠ loga + logb
  • Distribute coefficient in power rule
  • Domain of log: x > 0 always
← Back to Overview