Revise: Logarithmic and Exponential Functions

y=bˣ: range (0,∞), passes through (0,1), asymptote y=0.

2⁻²=0.25.

y=log_b(x): inverse of bˣ, domain (0,∞).

log₂(8)=3.

Vertical shift D also moves the asymptote.

3ˣ+6 → asymptote y=6.

d/dx[eˣ] = eˣ — its own derivative.

3eˣ → f'(0)=3.

d/dx[bˣ] = bˣ·ln(b).

2ˣ → f'(0)≈0.693.

d/dx[lnx] = 1/x.

5lnx → f'(2)=2.5.

d/dx[log_b x] = 1/(x ln b).

log₂(x) → f'(1)≈1.443.

bˣ is always strictly positive.

3²=9.

The log graph is the exponential graph reflected over y=x.

log₃(9)=2.

x−C shifts right; +D shifts up, moving the asymptote too.

2ˣ⁻¹+3: shifted right 1, up 3.

e is defined so this self-derivative property holds.

eˣ+5 → f'(0)=1.

Derived from bˣ = e^(x ln b) via the Chain Rule.

ln(e)=1 simplifies the formula.

Connects to the "missing case" of the power rule.

lnx → f'(4)=0.25.

Derived using the change-of-base identity.

ln(e)=1 simplifies to 1/x.