Logarithms and exponential functions

Question 1

Properties of logarithmic functions (7)

Answer 1

logᵦ(b) = 1
logᵦ(1) = 0
logᵦ(bⁿ) = n
logᵦ(xⁿ) = n * logᵦ(x)
logᵦ(ⁿ√x) = [1/n] * logᵦ(x)
logᵦ(uv) = logᵦ(u) + logᵦ(v)
logᵦ(u/v) = logᵦ(u) - logᵦ(v)
log(1/x) = -log(x)

Question 2

log(x) vs ln(x)

Answer 2

logₑ(x) = ln(x)

Note:
If no base is specified log(x) equals log₁₀(x)

Question 3

Transforming log equations

Answer 3

x = logᵦ(y) –> y = bˣ

Question 4

Modeling exponential relationships in a linear form

Answer 4

y = axⁿ
log(y) = n log(x) + log(a)

y = mx + c

Question 5

Determining the power law model

Answer 5

1. Transform equation by taking logs on both sides
2. Substitute (x₁,y₁) and (x₂,y₂) from 2 suitable points into the transformed log equation
3. Solve the simultaneous equations to find n
4. Substitute (x₁,y₁) and n into y = axⁿ to find a value.