College Algebra - Week 11

Exponentials and Logarithms

Exponential and logarithmic functions describe growth, decay, and scaling across every field of science, finance, and information theory. This week you explore the graphs and properties of exponential functions, derive logarithms as their inverses, master the three core logarithm laws, and solve exponential and logarithmic equations. These functions appear directly in calculus through derivatives and integrals.

Learning Objectives

- Evaluate and graph exponential functions f(x) = a^x and understand base requirements. - Identify exponential growth (base > 1) and exponential decay (0 < base < 1). - Define logarithm as the inverse of exponentiation and convert between forms. - Evaluate common logarithms (base 10) and natural logarithms (base e). - Apply the product, quotient, and power rules of logarithms. - Solve exponential equations by taking logarithms of both sides. - Solve logarithmic equations and check for extraneous solutions.

1. Exponential Functions: f(x) = a^x

An exponential function has the form f(x) = a^x where a > 0 and a ≠ 1. If a > 1: exponential growth. As x increases, f(x) grows rapidly. If 0 < a < 1: exponential decay. As x increases, f(x) decreases toward 0. Examples: f(x) = 2^x → f(0)=1, f(1)=2, f(2)=4, f(-1)=1/2 f(x) = (1/2)^x → f(0)=1, f(1)=1/2, f(2)=1/4 Key features: - Domain: all real numbers. - Range: (0, infinity). - Always passes through (0, 1). - Horizontal asymptote at y = 0.

2. The Natural Base e

The number e ≈ 2.71828... is the most important base in mathematics. The function f(x) = e^x appears in every natural growth/decay process: - Compound interest: A = Pe^(rt) - Radioactive decay: N = N_0 * e^(-kt) - Population growth: P = P_0 * e^(kt) The natural exponential f(x) = e^x has the special property that its derivative is itself — a fact that makes it essential in calculus. f(-1) ≈ 0.368, f(0) = 1, f(1) ≈ 2.718, f(2) ≈ 7.389

3. Logarithms: The Inverse of Exponents

The logarithm base a answers: 'What exponent do I need to raise a to in order to get x?' Definition: log_a(x) = y ⟺ a^y = x This is called the logarithmic form ↔ exponential form conversion. Examples: log_2(8) = 3 because 2^3 = 8. log_3(81) = 4 because 3^4 = 81. log_5(1) = 0 because 5^0 = 1. log_4(1/16) = -2 because 4^(-2) = 1/16. The common log (base 10): log(x) = log_10(x). The natural log (base e): ln(x) = log_e(x).

4. Converting Between Forms

Exponential form: a^y = x Logarithmic form: log_a(x) = y These say exactly the same thing in two different languages. Convert to log form: 5^3 = 125 → log_5(125) = 3 10^4 = 10000 → log(10000) = 4 e^2 = x → ln(x) = 2 Convert to exponential form: log_2(32) = 5 → 2^5 = 32 ln(y) = 3 → e^3 = y log(1000) = 3 → 10^3 = 1000

5. Properties of Logarithms

Three core laws (for a > 0, a ≠ 1, M > 0, N > 0): Product rule: log_a(MN) = log_a(M) + log_a(N) Quotient rule: log_a(M/N) = log_a(M) - log_a(N) Power rule: log_a(M^r) = r * log_a(M) Special values: log_a(1) = 0 (because a^0 = 1) log_a(a) = 1 (because a^1 = a) a^(log_a(x)) = x (inverse property) log_a(a^x) = x (inverse property)

6. Expanding and Condensing Logarithms

Expanding: break a single log into a sum/difference. Example: Expand log(x^2 * y^3 / z). = log(x^2) + log(y^3) - log(z) = 2*log(x) + 3*log(y) - log(z) Condensing: combine multiple logs into one. Example: Condense 3*ln(x) - 2*ln(y) + ln(4). = ln(x^3) - ln(y^2) + ln(4) = ln(4x^3/y^2)

7. The Change of Base Formula

To compute log_a(x) on a calculator (which usually has only log and ln): log_a(x) = log(x)/log(a) = ln(x)/ln(a) Example: Compute log_5(47). log_5(47) = log(47)/log(5) = 1.672/0.699 ≈ 2.391 Or: ln(47)/ln(5) ≈ 3.850/1.609 ≈ 2.391 Both give the same answer. Use whichever you prefer.

8. Solving Exponential Equations

Strategy 1: If both sides can be written as the same base, set exponents equal. 2^(x+3) = 16 → 2^(x+3) = 2^4 → x+3 = 4 → x = 1. Strategy 2: Take a logarithm of both sides when bases cannot be matched. Example: Solve 5^x = 30. ln(5^x) = ln(30) x*ln(5) = ln(30) x = ln(30)/ln(5) ≈ 2.113 Example: Solve 3^(2x-1) = 20. (2x-1)*ln(3) = ln(20) 2x - 1 = ln(20)/ln(3) ≈ 2.727 2x ≈ 3.727 x ≈ 1.863

9. Solving Logarithmic Equations

Strategy 1: Isolate the logarithm and convert to exponential form. Example: Solve log_2(x - 3) = 4. Convert: x - 3 = 2^4 = 16. So x = 19. Check: log_2(16) = 4. True. Strategy 2: When logs are on both sides with the same base, equate arguments. log(3x - 1) = log(x + 7) → 3x - 1 = x + 7 → x = 4. Always check: the argument of a log must be positive.

10. Extraneous Solutions in Logarithmic Equations

Example: Solve log(x) + log(x - 3) = 1. Condense: log(x(x-3)) = 1 Convert: x(x-3) = 10 x^2 - 3x - 10 = 0 (x - 5)(x + 2) = 0 x = 5 or x = -2. Check x = -2: log(-2) is undefined. EXTRANEOUS. Check x = 5: log(5) + log(2) = log(10) = 1. True. Solution: x = 5.

11. Compound Interest and Growth Models

Compound interest: A = P(1 + r/n)^(nt) Continuous compounding: A = Pe^(rt) Example: $5000 invested at 6% compounded continuously. How long to double? 10000 = 5000e^(0.06t) 2 = e^(0.06t) ln(2) = 0.06t t = ln(2)/0.06 ≈ 11.55 years. Half-life model: A = A_0 * e^(-kt). Given half-life, find k: (A_0/2) = A_0 * e^(-kT) → 1/2 = e^(-kT) → k = ln(2)/T.

12. Common Mistakes

- log(M + N) ≠ log(M) + log(N). The product rule adds INSIDE the log. - log(M/N) ≠ log(M)/log(N). Division outside the log is NOT the quotient rule. - ln(e^5) = 5, not e^5. The ln and e^x cancel. - Forgetting to check that logarithm arguments are positive after solving. - Confusing log base 10 (log) with natural log (ln). - Applying the power rule: log(x^2) = 2*log(x) only holds when x > 0.