Exponential Functions

• General form: \(y = a \cdot b^{kx} + d\)
• Domain: all real numbers
• Range: \((d, \infty)\) if \(a > 0\); \((−\infty, d)\) if \(a < 0\)
• Horizontal asymptote: \(y = d\)
• Key points: \((0, a + d)\), \((1/k - c, ab + d)\)

Logarithmic Functions

• Definition: \(\log_b(N) = e \iff b^e = N\), with \(b > 0\), \(b \neq 1\)
• Parent function: \(y = \log_b(x)\)
• Domain: \((0, \infty)\)
• Range: \((−\infty, \infty)\)
• Vertical asymptote: \(x = -c\)
• Key points: \((1/k - c, d)\), \((b/k - c, a + d)\)

Laws of Exponents

• \(b^m \times b^n = b^{m + n}\)
• \(b^m \div b^n = b^{m - n}\)
• \((b^m)^n = b^{m \cdot n}\)
• \((bc)^m = b^m \cdot c^m\)
• \(1 / b^m = b^{-m}\)

Properties of Logarithms

• \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• \(\log_b(M/N) = \log_b(M) - \log_b(N)\)
• \(\log_b(M^k) = k \cdot \log_b(M)\)
• Change of base: \(\log_b(M) = \frac{\log_a(M)}{\log_a(b)}\)

Relationship Between Exponents and Logs

• Exponential: \(b^e = N\)
• Logarithmic: \(\log_b(N) = e\)
• They are inverses: if \((x, y)\) is on exponential, then \((y, x)\) is on logarithm.

Solving Exponential and Logarithmic Equations

• Convert exponential to logarithmic form and vice versa.
• Combine log expressions using product, quotient, and power rules.
• Always check: log arguments must be greater than \(0\).

Word Problem Models

• Discrete growth: \(A = A_0(1 + r)^t\)
• Doubling time: \(t = \frac{\log(2)}{\log(1 + r)}\)
• Continuous growth/decay: \(A = A_0 e^{kt}\)
• Doubling with continuous growth: \(t = \frac{\ln(2)}{k}\)
• Half-life with continuous decay: \(t = \frac{\ln(1/2)}{k}\)

Asymptotes and Transformations

• Exponential parent: horizontal asymptote \(y = 0\); with transformations, asymptote is \(y = d\)
• Logarithmic parent: vertical asymptote \(x = 0\); with transformations, asymptote is \(x = -c\)
• Mapping rules:
• Exponential: \((x_p, y_p) \to (x_p/k - c, a y_p + d)\)
• Logarithmic: \((x_p, y_p) \to (x_p/k - c, a y_p + d)\)
• Inverse rule: exponential and logarithmic graphs are reflections across \(y = x\)