Revise: Logarithms

a × 10ⁿ, with 1 ≤ a < 10.

45,000,000 = 4.5 × 10⁷.

log_b(x) = y ⇔ bʸ = x.

log₂(8) = 3 ⇔ 2³ = 8.

log_b(MN) = log_b M + log_b N.

log₂(8×16) = 3+4 = 7.

log_b M = log_a M / log_a b.

log₇50 = log50/log7 ≈ 2.011.

ln(x) = log_e(x), e ≈ 2.71828.

ln(e⁵) = 5.

Move the decimal point until one nonzero digit remains; count the moves as n.

0.00032 = 3.2 × 10⁻⁴.

Multiply/divide coefficients and powers of 10 separately.

(3×10⁴)(2×10³) = 6×10⁷.

log_b(x) = y ⇔ bʸ = x — logs are exponents in disguise.

5² = 25 ⇔ log₅25 = 2.

log_b(x) = the exponent that turns b into x.

log₄64 = 3.

log_b(MN) = log_b M + log_b N.

log₃(9x) = 2 + log₃x.

log_b(M/N) = log_b M − log_b N.

log₂(x/8) = log₂x − 3.

log_b(Mᵖ) = p · log_b M.

log₂(4³) = 3·log₂4 = 6.

log_b M = log_a M / log_a b — needed for any base a calculator lacks.

log₃2 = log2/log3 ≈ 0.63.

log(x) with no base written means log₁₀(x).

log(100,000) = 5.

ln(x) = log_e(x); ln(eⁿ) = n.

ln(e³) = 3.

Isolate → combine into one log → convert to exponential form → solve → check the domain.

log₂x + log₂(x−2) = 3 → x = 4.