지수와 로그 함수: 세상을 움직이는 역함수 쌍
Why exp and log are inverses, the three log rules that solve everything, and the domain trap that sinks log equations.
Exponential and logarithm are inverses
If b > 0 and b ≠ 1:
bˣ = y ⟺ log_b(y) = x
So e^(ln a) = a and ln(eᵇ) = b. They undo each other.
The three log rules
- Product: log_b(MN) = log_b M + log_b N
- Quotient: log_b(M/N) = log_b M − log_b N
- Power: log_b(Mⁿ) = n · log_b M
And the identities: log_b(b) = 1, log_b(1) = 0, log_b(bᵏ) = k.
Solving exponential equations
If you can match bases, equate exponents: 2^(x+1) = 32 = 2⁵ → x + 1 = 5 → x = 4.
If you can't, take log of both sides: 3^x = 7 → x = ln(7)/ln(3) ≈ 1.77.
The log-equation domain trap
After solving ln(x) + ln(x − 3) = ln(10) → x = 5 or −2, the −2 is EXTRANEOUS because ln(−2) is undefined. Always check that the original log arguments are positive.
Growth and decay models
Continuous compounding / population growth: A(t) = A₀ · e^(rt). For r > 0 grows; r < 0 decays. Half-life formula: t½ = ln(2)/k.
Check yourself
Practice with real CBE questions
Try exp/log questions in Pre-Calc Sem A practice.