Exponential and Logarithmic Functions: Inverses That Run Your World
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.