Exponents vs Logarithms: The Basics

• Exponential: \(b^e = N\) means "base b to the power e equals N". Example: \(3^2 = 9\).
• Logarithm: \(\log_b(N) = e\) means "log base b of N equals exponent e". Example: \(\log_3(9) = 2\).
• Logs are the inverse of exponentials.

Domain & Range: Exponential vs Logarithmic

• Exponential \(y = b^x\) \((b > 0, b ≠ 1)\): Domain: all real \(x\); Range: \((0, ∞)\)
• Logarithmic \(y = \log_b(x)\): Domain: \((0, ∞)\); Range: all real \(y\)
• Inverse mapping: If exponential has \((x, y)\), log has \((y, x)\). So domain(log) = range(exp), range(log) = domain(exp).

Acceptable Bases & Growth/Decay

• Base \(b\) must be \(b > 0\) and \(b ≠ 1\).
• \(b > 1\): exponential growth; \(0 < b < 1\): exponential decay.
• Negative bases are not used for real-valued exponentials/logs in grade \(12\).

Triple A for Exponential Graphs

• Anchor point: \((0, 1)\) since \(b^0 = 1\)
• Asymptote: \(y = 0\) for parent \(y = b^x\) (shifts with vertical translation)
• Another point: \((1, b)\) since \(b^1 = b\)

Exponential Graph Behavior

• \(b > 1\): increasing (rises to right). Example: \(y = 2^x\)
• \(0 < b < 1\): decreasing (decay). Example: \(y=(1/2)^x\)
• Parent \(y = b^x\) has HA: \(y = 0\). For \(y = a b^{k(x + c)} + d\), HA is \(y = d\).

Transforming Exponential Graphs

• General form: \(y = a b^{k(x + c)} + d\)
• a: vertical stretch/compression, reflect \(x\)-axis if \(a < 0\)
• k: horizontal compression/stretch, reflect \(y\)-axis if \(k < 0\)
• c: horizontal shift (left \(+c\), right \(-c\))
• d: vertical shift (up \(+d\), down \(-d\))
• Mapping: \((0,1)\) → \((−c, a + d)\), \((1, b)\) → \((1/k − c, a b + d)\), HA: \(y = d\)

Graphing Logarithmic Functions

• Parent: \(y = \log_b(x)\)
• Anchor: \((1, 0)\) since \(\log_b(1) = 0\)
• Asymptote: \(x = 0\) (vertical)
• Another point: \((b, 1)\) since \(\log_b(b) = 1\)
• Transformed: \(y = a \log_b(k(x + c)) + d\), VA: \(x = -c\)
• Mapping: \((1,0)\) → \((1/k − c, d)\), \((b,1)\) → \((b/k − c, a + d)\)

Estimating Logs (1 Decimal Place)

• Change of base: \(\log_b(x) = \frac{\ln(x)}{\ln(b)}\) or \(\frac{\log_{10}(x)}{\log_{10}(b)}\)
• No calculator: find two nearby powers of \(b\), interpolate
• Example: \(\log_2(10)\) is between \(3\) and \(4\) (since \(2^3 = 8\), \(2^4 = 16\)), estimate \(≈ 3.3\)

Logarithm Properties

• Product: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient: \(\log_b(M/N) = \log_b(M) - \log_b(N)\)
• Power: \(\log_b(M^p) = p \log_b(M)\)
• Change of base: \(\log_b(A) = \frac{\ln(A)}{\ln(b)}\)
• Equality: \(\log_b(M) = \log_b(N) \implies M = N\) (\(M\), \(N > 0\))
• Exponential equality: \(b^u = b^v \implies u = v\) \((b > 0, b ≠ 1)\)

Solving Exponential & Log Equations

• Exponential to log: \(b^x = A \implies x = \log_b(A) = \frac{\ln(A)}{\ln(b)}\)
• Log to exponential: \(\log_b(E) = v \implies E = b^v\)
• Combine logs: \(\log_b(M) + \log_b(N) = k \implies MN = b^k\)
• Exponential with linear exponent: \(A b^{kt} = C \implies t = \frac{1}{k} \log_b(C/A)\)
• Always check domain: log arguments > \(0\); exponentials, check range if needed

Word Problems: Growth, Decay, and Models

• Discrete: \(A(t) = A_0 b^t\), doubling time: \(T_{double} = \log_b(2)\)
• Continuous: \(A(t) = A_0 e^{kt}\), doubling: \(T_{double} = \frac{\ln(2)}{k}\), half-life: \(T_{half} = \frac{-\ln(2)}{k}\)
• Translate words to formula, substitute, solve for unknown using logs, check units, interpret

Asymptotes & Shifts

• Exponential parent \(y = b^x\): HA \(y = 0\). For \(y = a b^{k(x + c)} + d\), HA is \(y = d\)
• Log parent \(y = \log_b(x)\): VA \(x = 0\). For \(y = a \log_b(k(x + c)) + d\), VA is \(x = -c\)

Quick Mapping Rules

• Exponential: (\(x_p, y_p\)) on parent \(y_p = b^{x_p}\), transformed: \(x = x_p/k - c\), \(y = a y_p + d\)
• Log: \((x_p, y_p)\) on parent \(y_p = \log_b(x_p)\), transformed: \(x = x_p/k - c\), \(y = a y_p + d\)
• Inverse: if exponential has \((x, y)\), log has \((y, x)\) (reflect across \(y = x\))

Common Exam Traps & Tips

• State domain restrictions before solving log equations (argument \(> 0\))
• For exponentials, use logs to solve; for logs, convert to exponentials
• Use exact forms (ln, log, fractions), round only final answers
• Label asymptotes and anchor points on graphs
• For word problems, write the model, identify units, show algebraic steps